Aggregation of Preferences for Skewed Asset Returns∗
Fousseni Chabi-Yo
Fisher College of Business, Ohio State University
Dietmar P.J. Leisen
Gutenberg School of Management and Economics, University of Mainz
Eric Renault
Department of Economics, Brown University
This version: June 5, 2014
Abstract
This paper characterizes the equilibrium demand and risk premiums in the presence of skewness risk. In a model with a single time period, we extend the classical mean-variance two-fund
separation theorem to a three-fund separation theorem. The additional fund is the skewness
portfolio, i.e. a portfolio that gives the optimal hedge of the squared market return; it contributes to the skewness risk premium through co-variation with the squared market return and
supports a stochastic discount factor that is quadratic in the market return. When the skewness
portfolio does not replicate the squared market return, a tracking error appears; this tracking
error contributes to risk premiums through kurtosis and pentosis risk. In addition to the common
powers of market returns, this tracking error shows up in stochastic discount factors as priced
factors that are products of the tracking error and market returns. Extending the one-period
to a two-period framework introduces an additional volatility-based risk factor in the stochastic
discount factor as well as a hedging demand in individual asset allocations due to volatility risk.
Keywords
skewness, portfolio, aggregation, stochastic discount factor, polynomials of pricing factors, market return
JEL Classification
D52, D53, G11, G12
∗
We thank Kenneth Judd for very helpful conversations. Please address correspondence to: Dietmar Leisen,
Faculty of Law and Economics, University of Mainz, 55099 Mainz, Germany; Phone: ++49-6131-3925542; Fax:
++49-6131-3923782; Email: leisen@uni-mainz.de.
1
1
Introduction
Empirical studies documented that securities’ returns are not normally distributed; the seminal
work of Harvey and Siddique (2000) showed that skewness risk is an important component in the
risk-premium. This renewed interest in the compensation of skewness risks and led to an active
literature stream1 . This stream typically assumes that the aggregation of preferences leads to
stochastic discount factors that are polynomials in the market return, conditional on the current
information set. However, it is well known that the aggregation of preferences in incomplete
markets leads to a representative agent where the weight of each individual is stochastic; thus,
the stochastic discount factor may actually depend on individual securities due to unspanned
powers of the market return. We study in detail the aggregation of preferences with skewed
returns.
Our paper proceeds through two steps. Although the focus of our analysis is on aggregation
of (heterogeneous) investors, our first step studies the case of a representative agent. We show
that the common practice of using polynomials in the market return is warranted as long as the
representative investor is myopic, i.e. as long as she only cares about investing over the next time
period; in line with this, we link our results to the beta pricing relationships proposed by Harvey
and Siddique (2000). However, when the representative investor derives utility from terminal
wealth by investing over two time periods, we find that that assets’ risk premiums over the first
period also depend on its (intertemporal) co-variation with the (conditional) market variance
over the second period; this can be seen as a form of co-market volatility. This additional covariation extends the usual variance-skewness pricing relationships in the literature and shows
up in the stochastic discount factor as a priced factor that is in addition to polynomials of the
market return.
Our major second step studies equilibrium risk premiums and demand with heterogeneous
investors. With myopic investors, i.e. investors that care only about the current time period, we
derive individual demand and prove a three-fund separation theorem: agents hold the risk-free
asset, the market portfolio and a new, so-called skewness portfolio, in proportions that reflect
1
For example, Dittmar (2002) and Barone-Adesi et al. (2004) study how skewness risk is priced; Chung et al.
(2006) test whether Fama-French factors proxy for skewness and higher moments; Engle (2011) links the skewness
that shows up in asymmetric volatility models to systemic risk; Boyer and Vorkink (2013) examine the impact of
skewness preference on option prices.
2
their preferences for variance and skewness risk. The skewness portfolio is the portfolio that
provides the optimal hedge to the squared market return. We show that an asset’s skewness risk
is priced as long as the co-skewness of this asset with the market portfolio as well as the aggregate
skew-tolerance do not vanish. The equilibrium contribution to the risk-premium of individual
stocks is driven by their covariance with the squared market return; it matches formally the
risk-premium contribution in the single period, representative investor case and supports the
common practice of using a stochastic discount factor (SDF) that is quadratic in the market
return.
To shed further light on the co-leverage (market volatility) factor that was priced in the
two period case with a representative investor, we extend the previous analysis and study the
equilibrium allocation with heterogeneous investors that maximize terminal wealth derived from
investing over two periods of time. We prove a four-fund separation theorem over the first period:
in addition to the risk-free asset, the market and the skewness portfolio, investors hold a new,
so-called leverage portfolio, in proportions that reflect their preferences for variance/skewness
risk and for wealth induced (intertemporal) changes in the risk tolerance. The leverage portfolio
is the portfolio (over the first time period) that provides the optimal hedge to the (conditional)
variance of the market return over the second period; investors’ holdings can be interpreted as a
form of the well-known intertemporal hedge demand term that is common in portfolio selection
models. The equilibrium contribution to the risk-premium depends, among others, on the covariation of individual stocks with the (conditional) market variance over the next period. It
matches formally the risk-premium contribution in the single period, representative investor case;
this additional co-variation with market volatility extends the usual variance-skewness pricing
relationships in the literature.
The remainder of our analysis focuses on the single period case with heterogeneous investors.
A tracking error shows up in hedging the squared market return with the available securities;
beyond average skew-tolerance, this leads to an additional lower order contribution to the risk premium through kurtosis due to the so-called cross-sectional variance of investor’s skew-tolerances.
Put differently, there may be an additional priced factor that contributes to skewness risk: the
product of the market return with the tracking error in the squared market return. Another
tracking error shows up in hedging the cubed market return with the available securities; beyond
3
average skew-tolerance, this leads to an additional lower order contribution to the risk premium
through pentosis (the fifth moment) due to the cross-sectional variance and the so-called crosssectional skewness of investor’s skew-tolerances.
Our paper makes several contributions. First, we provide a structural analysis of individual
and aggregate2 demand functions, as well as equilibrium risk premiums and associated portfolio holdings. We revisit the seminal contribution of Samuelson (1970), see also the small risk
analysis in Gollier (2001), but depart from it in two directions: for one, we allow for a richer
pattern of risk premiums in order to analyze the pricing of skewness risk; in addition, we do not
intend to interpret his small noise expansion parameter as we are interested only in structural
properties of risk-premiums and portfolio holdings. Thereby we confirm Samuelson (1970) for
myopic investors: the classical CAPM two fund separation theorem continues to hold as a first
approximation; in addition we show that agents add the skewness portfolio in a second order approximation and that higher order approximations lead to lower order contributions of skewness
risk through tracking errors.
Second, we point out in two-period extension that the intertemporal hedge terms of nonmyopic investors will affect demand through the leverage portfolio and risk-premiums through
an additional priced factor; the latter shows up as a co-market volatility effect. This fits into a
recent literature that considers the feedback effect on pricing of (intertemporal) changes in the
investment opportunity set: for example, in presence of long dated assets, i.e. assets with payoffs
that are far in the future, Martin (2012)3 uses the SDF over multiple periods to show that the
pricing of a broad class of these assets is driven by the possibility of extraordinarily bad news.
Finally, we clarify the common practice of studying SDF that are polynomials in the market
return. Most empirical studies looked at skewness extensions of the CAPM which add the
squared market return as a factor; they justify this extension on ad-hoc Taylor-series expansions
for agents’ utility functions that are truncated at the third-order term, see, e.g., Kraus and
Litzenberger (1976), Barone-Adesi (1985), Dittmar (2002). We provide a rigorous foundation for
2
Structural properties of the market demand functions have been studied, e.g. by Hildenbrand (1983) and
Grandmont (1992). We focus on a portfolio choice problem.
3
An example in Martin (2012), page 349, Figure 1 illustrates this point when the investment horizon is
t=250 years. Our framework consider only implication of heterogeneity in one-period (when t=0,1) and twoperiod models (when t=0,1,2). To shed light on Martin’s finding within our framework, one has to look at the
implication of our model for many more than two time-periods. This is beyond the scope of this paper and merits
scrutiny in future research.
4
this common practice and show that it is warranted with myopic investors in complete markets,
that is when investors only care about one single time-period and the squared market return
can be hedged perfectly. When investors are not myopic or markets are incomplete, however, we
show that several additional priced factors come up that depend on the cross-sectional variance
and skewness of investors’ skew-tolerances.
Our structural result in single-period cases unifies several theoretical insights. First of all,
our results match previous ones about incomplete markets that investors’ heterogeneity and in
particular the cross-sectional variance of investors’ characteristics may matter in equilibrium
pricing, see, e.g., Constantinides and Duffie (1996). In addition, we show that aggregation of
skewness risk for pricing is far more complicated than thought: the standard approach of studying polynomials of market returns (based on a postulated representative agent) misses several
additional, priced factors that are products of market returns with tracking portfolios. Finally,
it is well known about incomplete markets that the utility functions aggregate into a single socalled representative agent through stochastic weights, see, e.g. Magill and Quinzii (1996); our
structural result therefore has the interpretation that these stochastic weights matter for pricing.
Our structural analysis also provides an explanation for recent empirical evidence: Mitton and
Vorkink (2007) and Boyer and Vorkink (2013) document for stocks and options, respectively,
that more than co-skewness with the market is priced in stock returns; our unspanned factors
provide a theoretical foundation for this.
The outline of our paper is as follows. The next section discusses the general framework and
relevant concepts. The third section studies the representative investor case, while the following
sections study heterogeneous investors. Section 4 discusses demand and the portfolio separation
theorem in the presence of skewness risk as well as the pricing of skewness risk. Finally, section 5
shows that the aggregation in incomplete markets leads to several risk factors that are unspanned
by polynomial SDFs in the market return. The paper concludes with section 6. All proofs are
postponed to the appendix: appendix A considers the proofs in the single-period case; appendix
B provides the proofs in the two-period case and is intended as an online appendix.
5
2
General return framework
This section introduces the general framework and the relevant concepts for understanding the
market prices of higher order moments. Throughout, we consider a scale-location model for the
vector R = (Ri )1≤i≤n of returns on the n risky assets of interest on a given period of time. More
precisely, there exists a random vector Y = (Yi )1≤i≤n with zero mean vector under some given
joint probability distribution, such that for σ > 0:
R = E(R) + σY,
(1)
where E(R) is the expectation of R. We always assume that the vector Y admits finite moments
at any required order and that the variance matrix V ar(Y ) is positive and symmetric. Note that
return volatilities are determined by the scale parameter σ through
V ar(R) = σ 2 Σ, with Σ = V ar(Y ).
Since our interest focuses on (the cross section of) risk premiums, we introduce a risk free
return Rf and decompose expected returns as follows:
E(Ri ) = Rf + πi (σ), i = 1, ..., n;
here, πi (σ) denotes the risk premium on asset i. This risk premium must obviously depend on
the scale of risk σ and we will always assume that it is a smooth function of σ. In particular,
when σ goes to zero, all asset volatilities go to zero and we expect risk premiums doing the same.
Extending by continuity the risk premium function, we will assume throughout that:
πi (0) = 0, i = 1, ..., n.
We did not define our financial markets for σ = 0, since we would end up with (n+1) risk-free
assets. However, it makes sense to study the behavior of the risk premium function in a right
neighborhood of zero, involving (by continuity extension) the value of the function πi (·) as well
as of its derivatives at σ = 0. Smoothness properties of risk premiums functions will be implied
by assumed smoothness properties of investors’ utility functions at stake. Let us assume for the
moment that some investor with utility function u chooses her portfolio as solution of the first
order conditions
E [u′ (W (σ))(Ri − Rf )] = 0, for i = 1, ..., n.
6
(2)
These optimality conditions (2) are very general when the random variable W (σ) stands
for some wealth level produced by the portfolio choice of interest. These equations may stem
from both static or dynamic portfolio optimization. We do not want to make this explicit at
this stage, it is only important to assume that all randomness in W (σ) is produced by asset
volatilities and that it is erased when the scale of risk σ goes to zero. It is worth realizing that,
up to required smoothness and integrability conditions, this very general setting already implies
that risk premiums have a zero (right-hand) derivative at σ = 0:
Lemma 1 If u′ (W (0)) is a deterministic non-zero variable, we have πi′ (0) = 0 for all i = 1, ..., n.
Lemma 1 is consistent with the general result (see e.g. Gollier (2001), p. 56) that the
optimal share of wealth to invest in a risky asset is approximately proportional to the ratio of
the expectation and variance of the excess return. Lemma 1 implies that risk aversion considered
in this paper has a second-order nature, as defined by Segal and Spivak (1990). (First order risk
aversion is beyond the scope of this paper.)
Based on Lemma 1, the scale parameter σ allows us to describe the pattern of risk premiums
from their series expansions:
πi (σ) = E(Ri ) − Rf =
∞
∑
aij σ j .
(3)
j=2
Samuelson (1970) assumes that the above expansion involves only one term, i.e. he sets aij = 0
for all j > 2; this, however, appears overly restrictive for a thorough analysis of the price of risk
coming from higher order moments beyond variance. In addition, Samuelson (1970) is interested
in small scale parameters σ; as he intends to motivate the use of mean-variance analysis, he
also puts forward an interpretation in continuous time, where the scale parameter σ corresponds
to a small interval of time ∆t. No such restrictions are involved in our setting defined by (1)
and (3). This setting is actually general, as it only maintains smoothness assumptions about
investors’ preferences that are necessary to ensure that risk premiums are analytical functions of
the scale parameter σ. We will actually set the focus in this paper on the first four coefficients
ai2 , ai3 , ai4 and ai5 , so that we even do not really assume that risk premiums are analytical
functions. There are then two interpretations of our results: either we adopt the assumption of
analytical functions and describe general properties of the market; or we may consider, under less
7
restrictive assumptions, that our results are accurate descriptions only for small σ, see Samuelson
(1970) and more recently Judd and Guu (2001).
Since our analysis focuses on the role of investors’ preferences4 , we assume throughout that
only investor’s demand (may) depend on the scale of risk σ but that the supply of financial assets
is given independently of σ. In other words, a market portfolio is given by a vector ξ of shares
of total wealth invested in the n risky assets. Since there is also a risk-free asset and no short
sale constraints, the vector ξ can be any vector in Rn with strictly positive entries. It defines the
market return
RM = ξ ⊥ R,
(4)
where we denoted by ξ ⊥ the transpose of the vector ξ. Throughout this paper we will refer by
the superscript ⊥ to the transpose of a vector.
From the extant literature on pricing of higher order moments (see Chung et al. (2006) and
references therein), we expect the risk premium on any risky asset i = 1, ..., n, to be determined
by the various contributions of this asset to market variance, skewness, kurtosis, pentosis, etc.
To study this, we adopt:
Definition 1 For asset i = 1, ..., n, the systematic co-moments are defined as follows:
[
]
1
⊥
Cov
[R
,
(R
−
E(R
))]
=
Cov
Y
,
(ξ
Y
)
,
i
M
M
i
σ2
[
]
[
]
1
2
⊥
2
Cov
R
,
(R
−
E(R
))
=
Cov
Y
,
(ξ
Y
)
,
=
i
M
M
i
σ3
[
]
[
]
1
3
⊥
3
=
Cov
R
,
(R
−
E(R
))
=
Cov
Y
,
(ξ
Y
)
,
i
M
M
i
σ4
[
]
[
]
1
4
⊥
4
Cov
R
,
(R
−
E(R
))
=
Cov
Y
,
(ξ
Y
)
.
=
i
M
M
i
σ5
bi,M =
ci,M
di,M
fi,M
We refer to bi,M , ci,M , di,M and fi,M as the market beta, the market co-skewness, the market
co-kurtosis and the market pentosis, respectively.
By comparison with standard notations, the definition of market moments and the asset comoments standardizes by the scale of risk; thus they do not depend on σ. This standardization
simplifies the presentation but does not change the main messages of the paper.
4
Our focus is on the demand side of financial markets and should be completed when the offer of financial
assets is also be seen as a function of the scale parameter σ. However, such an extension would simply complicate
notation without modifying the economic interpretation of results.
8
Note also that we set the focus on centered systematic co-moments and do not standardize by
variance; this is different from, e.g., Chung et al. (2006). Our convention allows us to interpret
the co-moments as shares of asset i in the aggregate corresponding moment namely respectively
market variance VM , skewness SM , kurtosis KM and pentosis PM :
n
∑
ξi bi,M =
[ ⊥ ]
1
V
ar
[R
]
=
V
ar
(ξ Y ) = VM ,
M
σ2
ξi ci,M =
]
[
]
1 [
E (RM − E(RM ))3 = E (ξ ⊥ Y )3 = SM ,
3
σ
ξi di,M =
]
[ ⊥ 4]
1 [
4
E
(R
−
E(R
))
=
E
(ξ Y ) = KM ,
M
M
σ4
ξi fi,M =
]
[ ⊥ 5]
1 [
5
E
(R
−
E(R
))
=
E
(ξ Y ) = PM .
M
M
σ5
i=1
n
∑
i=1
n
∑
i=1
n
∑
i=1
From an economic perspective, it is worth pointing out that even if the market skewness or
pentosis is zero, some specific asset may have a non-zero systematic component and that this
may play a role in the determination of its market price.
Chung et al. (2006) study asset pricing implications of a model that “does not stop at the
second moment” and considers up to the fourth moment. They stress that “there is no reason to
stop with the fourth moment”, i.e. even higher order moment like pentosis may matter for asset
pricing. They base their argument on the remark that “variance, skewness and kurtosis tell us
something about the tails of the distribution, but they fall short of specifying the tail precisely”.
In this paper, our main interest is in the price of skewness; we will argue that when considering
investors with heterogeneous preferences for skewness, an asset pricing model that does not want
“to stop with the second moment” must take also into account some specific pricing factors due
to preference heterogeneity. To put it differently, we will argue that the simple fact that investors
care for higher order moments implies that they need to track not only the market return, but
also the pricing factor (RM − E(RM ))2 that shows up through market co-skewness. We will see
that this has a significant pricing impact in case of market incompleteness (there is no perfect
hedge of the squared market return) jointly with investor heterogeneity. To see this, we will
study terms in the series expansion (3) as far as ai4 or ai5 and the associated pricing of market
co-kurtosis and market co-pentosis.
9
3
Pricing kernels and moment preferences in representative agent analysis
This section studies the pricing of (higher order) moments without taking into account complications due to preference heterogeneity: throughout this section, we assume that the representative
investor has a known preference structure. This assumption will be questioned in the following
sections through explicit discussions of aggregation issues.
Ultimately, our analysis with heterogeneous investors implies that the representative investor
paradigm falls short of properly characterizing the impact of preferences for skewness on financial
asset prices. However, before carrying out this analysis we first recall in this section the classical
picture of risk premiums assigned to systematic co-moments in the case of a representative
investor. The pricing formulas in the literature that result from a Taylor expansion of the utility
function of a representative agent (see e.g. Harvey and Siddique (2000) and Dittmar (2002)) will
find a more general interpretation in the context of our series expansions (3).
3.1
Case of a representative investor over one period
This subsection starts our analysis of pricing by studying a single period of time. Let us consider
a representative investor who maximizes expected utility E[u(W )] of her terminal wealth W
obtained by investing her initial wealth q in a portfolio. Throughout, the portfolio is defined in
terms of shares of wealth invested θ = (θi )1≤i≤n ; this gives terminal wealth:
[
]
n
∑
W (σ) = q Rf +
θi (Ri − Rf ) .
(5)
i=1
The demand θ of the representative investor for risky assets solves the first order conditions
(2) of her expected utility maximization where the wealth level W (σ) is induced by the portfolio
choice according to (5). Since the representative investor must hold the market portfolio, the
first order conditions with θ = ξ determine the equilibrium pricing of the financial assets, i.e.
the coefficients aij , j ≥ 2 of the risk premium expansion (3); these coefficients solve for all levels
10
σ of risk:
E [u′ (W (σ))(Ri (σ) − Rf )] = 0 for i = 1, ..., n,
[
]
n
∑
subject to W (σ) = q Rf +
ξi (Ri (σ) − Rf ) ,
(6)
i=1
where Ri (σ) − Rf = πi (σ) + σYi for i = 1, ..., n.
We are then able to prove:
Theorem 1 The equilibrium risk premium πi (σ) = E(Ri ) − Rf =
∑∞
j=2
aij σ j fulfills
1
ρ
κ
1 − 3ρ
bi,M , ai3 = − 2 ci,M , ai4 = 3 di,M +
VM bi,M ,
τ
τ
τ
τ3
χ
3κ − ρ(1 − ρ)
κ − ρ(1 − 2ρ)
= − 4 fi,M +
V
c
+
SM bi,M ,
M
i,M
τ
τ4
τ4
ai2 =
and ai5
where
u′ (qRf ) u[3] (qRf )
u′ (qRf )
,
ρ
=
,
qu′′ (qRf )
2[u′′ (qRf )]2
[u′ (qRf )]2 u[4] (qRf )
[u′ (qRf )]3 u[5] (qRf )
κ =
, χ=
.
6[u′′ (qRf )]3
24[u′′ (qRf )]4
τ = −
Theorem 1 puts forward the preference parameters τ, ρ, κ, χ that characterize the representative investors’ concerns for higher order moments. Following Scott and Horvath (1980), we
expect investors to have a negative preference for even moments and a positive preference for
odd ones. Caballe and Pomansky (1996) have shown that this property is displayed by utility
functions exhibiting “mixed risk aversion”, that is having all odd derivatives positive and all
even derivatives negative. As noted by Pratt and Zeckhauser (1987), the latter property means
that utility functions have completely monotone first derivatives, a property known to correspond to Laplace transforms of probability distributions on the positive real line. Eeckhoudt
and Schlesinger (2006) have provided some theoretical underpinnings for this property in terms
of “risk apportionment”. We will then rely upon this literature to maintain throughout the
following assumptions:
u[k] (qRf ) ≤ 0 for k even , and u[k] (qRf ) ≥ 0 for k odd .
Under this assumption, all the preference parameters τ, ρ, κ, χ of Theorem 1 are non-negative;
since they contain successively a higher order derivative of the representative investor’s utility
11
function u, we call them risk tolerance, skew tolerance, kurtosis tolerance and pentosis tolerance,
respectively5 . Theorem 1 shows that when preference parameters ρ, κ, χ for higher order moments
are rescaled by the right power of risk tolerance, they determine the risk premium contribution
of the corresponding market co-moment.
It was the key insight of Samuelson (1970), that an approximation of the risk premium
(E(Ri ) − Rf ) by the first term ai2 σ 2 would provide an asset pricing model that matches the
traditional Sharpe-Lintner CAPM. Thus, to understand the price of skewness risk, it is necessary
to study at least an expansion with one order higher. It has been well known since Kraus and
Litzenberger (1976) that it is the co-skewness ci,M of an asset and not its skewness that determines
its risk premium. Theorem 1 matches that intuition.
It is helpful to compare our result in theorem 1 with the usual approach of Taylor expansions
of the representative investor’s marginal utility (see e.g. Dittmar (2002)). To study higher terms
let us now consider the risk premium up to the fourth term based on theorem 1:
ai2 σ 2 + ai3 σ 3 + ai4 σ 4 + ai5 σ 5
(
)
1 1 − 3ρ
κ − ρ(1 − 2ρ)
3
=
+
V ar(RM ) +
E[(RM − E(RM )) ] Cov [Ri , (RM − E(RM ))]
τ
τ3
τ4
)
(
[
]
3κ − ρ(1 − ρ)
ρ
2
V
ar(R
)
Cov
R
,
(R
−
E(R
))
+ − 2+
M
i
M
M
τ
τ4
[
] χ
[
]
κ
+ 3 Cov Ri , (RM − E(RM ))3 − 4 Cov Ri , (RM − E(RM ))4 .
τ
τ
Note that the pricing impact of market beta Cov [Ri , (RM − E(RM ))] is driven not only by
the inverse of risk tolerance τ but also V ar(RM ) together with the skew tolerance and the third
moment of the market return together with kurtosis tolerance matter; a similar statement can
be made for co-skewness. However, along the lines of a Taylor series expansion of the stochastic
discount factor (see e.g. Dittmar (2002)) we would expect that the pricing impact of co-moments
of an asset i with higher order market terms (RM − E(RM ))j (j = 1, 2, 3, 4) is driven only by
the preference parameters τ, ρ, κ, χ, respectively. (This is akin to expect that only the terms
5
Skewness tolerance is related to the prudence concept of Kimball (1990); kurtosis tolerance is related to the
temperance concept of Kimball (1992) and to risk vulnerability of the utility function in the presence of background
risks, see Gollier (2001). Eeckhoudt and Schlesinger (2006) study lotteries with the property that prudence and
temperance can be fully characterized by a preference relation over these lotteries. Pentosis tolerance is driven
by the fifth derivative of the utility function. It is related to the concept of edginess that Lajeri-Chaherli (2004)
introduced to study decision making in the presence of background risks. Eeckhoudt (2012) discusses the sign of
various derivatives (third, fourth, fifth) of the utility function.
12
κ
d
τ 3 i,M
and − τχ4 fi,M determine ai4 and ai5 , respectively.) The proof of theorem 1 shows that this
omission is akin to making only the series expansion of the marginal utility while forgetting that
the optimality condition (whose expansion is called for) actually involves the product of marginal
utility times net return: this is akin to overlooking that (higher order) moments of the product
of marginal utility with risky assets’ returns are characterized through all products of suitable
(lower order) moments of marginal utility and of suitable (lower order) moments of risky assets.
For illustration of the economic impact, let us consider for a moment the case of a utility
function with constant relative risk aversion:
u′ (w) = w−α , α > 0.
For this utility function we have:
ρ=
1
1
1
+
>
2 2α
2
Theorem 1 then implies that for an asset i with positive market beta and positive market cokurtosis, the risk premium for large co-kurtosis is smaller than expected in the Taylor series
approach (see, e.g. Dittmar (2002)), since it overlooks the correction term proportional to
1 − 3ρ < 0. A similar comment could be made about the risk premium for negative co-pentosis.
3.2
Case of a representative investor over two periods
This subsection continues to consider a representative investor who maximizes expected utility
E[u(W )] derived from the terminal wealth W that she obtains by investing some initial wealth
q in a portfolio. The difference to the previous subsection is that we take into account that
this terminal wealth may be the result of a dynamic investment strategy over several consecutive
periods. For sake of expositional simplicity, we consider only two periods with constant risk-free6
rate Rf = Rf,0 = Rf,1 : the first period runs from date 0 to date 1, the second period runs from
date 1 to date 2. In other words, the dynamic portfolio of the representative investor is defined
in terms of shares of wealth invested θ = (θi0 , θi1 )1≤i≤n to give terminal wealth:
]
]
[
[
n
n
∑
∑
0
1
θi (Ri,1 − Rf ) .
W2 = W1 · Rf +
θi (Ri,2 − Rf ) , W1 = q · Rf +
(7)
i=1
i=1
6
Allowing the interest rate to vary would lead to additional terms that characterize the ratio. That would not
affect our results qualitatively but would considerably complicate our notation.
13
The second period portfolio holdings may depend on the realization of returns over the first
period, but for simplicity of exposition we do not write out this dependence explicitly throughout
this subsection. Throughout, we consider a scale-location model for the vector of returns that is
defined recursively over consecutive periods:
R(1) = (Ri,1 )1≤i≤n = E(R(1) ) + σY(1) , with E(Y(1) ) = 0,
(
)
(
)
and R(2) = (Ri,2 )1≤i≤n = E R(2) Y(1) + σY(2) , with E Y(2) Y(1) = 0.
(8)
(9)
Note that we assume without loss of generality (up to a proper rescaling of the two random
vectors Y(1) and Y(2) ) that the scale parameter σ is the same in the two periods.
As in the previous subsection, we assume that the supply of risky assets is independent of σ.
At the beginning of the first period, the supply is given by a vector ξ of shares of total (date 0)
wealth q invested in the n risky assets. Similarly, we denote ξ 1 = (ξi1 )1≤i≤n the vector of total
date 1 wealth invested in the risky assets at the beginning of the second period. These define
the market returns RM,1 and RM,2 over both periods by setting
RM,1 = ξ ⊥ E(R(1) ) + σξ ⊥ Y(1) ,
(
)
RM,2 = ξ 1⊥ E R(2) Y(1) + σξ 1⊥ Y(2) .
It is important to note that the date 1 vector of invested wealth ξ 1 depends on the relative prices
of risky assets and that these (may) change over the first period. We assume that the total
supply of asset i is given over the two consecutive periods through the quantities xi,0 and xi,1 of
units of asset i; then we can write
ξi1 = ξi
Ri,1 xi,1
.
RM,1 xi,0
For sake of expositional simplicity, we assume:
xi,1 =
RM,1
xi,0 .
Ri,1
This simplifying assumption implies ξ 1 = ξ; in other words, we impose some neutrality in the
public offering of asset i: it ensures the invariance (with respect to both state variables Y and
scale of risk σ) of the market portfolio between the two dates in terms of shares of invested wealth.
Our simplifying assumption could be relaxed at the cost of more complicated pricing formulas
14
without changing the substance of the pricing of skewness that is the focus of our interest. We
can then write
RM,2 = ξ ⊥ E(R(2) |Y(1) ) + σξ ⊥ Y(2) .
Based on Lemma 1, the scale parameter σ allows us to describe the pattern of risk premiums
from their series expansions:
(
)
πi,1 (σ) = E Ri,2 | Y(1) − Rf =
∞
∑
a1ij σ j
and πi,0 (σ) = E (Ri,1 ) − Rf =
∞
∑
a0ij σ j .
(10)
j=2
j=2
At date 0, the representative investor faces the following dynamic optimization program:
max
E[u(W2 (σ))] = max
E[J(W1 (σ))],
θ0 ,θ1
θ0
]
where J(W1 (σ)) = max
E[u(W
2 (σ)) Y(1) ,
θ1
[
]
n
∑
W2 (σ) = W1 (σ) Rf +
θi1 (πi,1 (σ) + σYi,2 ) ,
[
]
i=1
W1 (σ) = q Rf +
n
∑
θi0 (πi,0 (σ) + σYi,1 ) .
i=1
Using the date 1 indirect utility function J defined above and the law of iterated expectations, we
can then write the investor’s first order conditions for date 0 portfolio optimization for i = 1, . . . , n
as
[
]
[
]
∂J ∂W1
∂W2
E u (W2 (σ))
(πi,0 (σ) + σYi,1 ) = E
= 0,
∂W1
∂W1 ∂θi0
′
(11)
where the terminal wealth level W2 = W2 (σ) is induced by the portfolio choice according to (7).
This describes the date 0 first order conditions.
We are solving this through backward induction. At date 1, given the observed realization
of the state vector Y(1) , the representative investor is faced with a given supply ξ of risky assets
and return prospects whose randomness is defined by the scale parameter σ and the conditional
probability distribution of Y(2) given Y(1) . We can then apply the one-period market results derived in the former subsection to conclude that the (conditional) date 1 equilibrium risk premium
for each asset i = 1, ..., n in the characterization of equation (10) is
a1i2 =
1
b1 ,
τ1 (qRM,1 ) i,M
15
a1i3 = −
ρ1 (qRM,1 ) 1
c .
τ12 (qRM,1 ) i,M
(12)
Here, preference parameters are defined through the date 1 risk tolerance and skew tolerance
functions
τ1 (w) = −
u′ (wRf )
,
wu′′ (wRf )
ρ1 (w) =
u′ (wRf ) u[3] (wRf )
.
2[u′′ (wRf )]2
(13)
Note that these are analogous to the risk and skew tolerances that we used in the single
period case, see theorem 1. Seen from date 0, these functions will be evaluated at qRM and so
they are random variables. Our approach allows us to view them through a series expansion in
the scale parameter and therefore we introduce the date 0 preference parameter
γ0 = q
∂τ1
(qRf ).
∂w
(14)
Note that τ1 /Rf is equal to the common (relative) risk tolerance function, evaluated at wRf .
Therefore, unless preferences are described through the so-called constant relative risk aversion,
the function τ1 will not be constant and wealth changes over the first period will affect the
investor’s risk tolerance at date 1. The term γ0 describes the first order change in the risk
tolerance function τ1 due to changes in investor’s wealth. Therefore, we call this preference
parameter the wealth tolerance.
Date 1 (conditional) systematic co-moments are defined here analogous to definition 1 by7
[
]
1
⊥
Y(1) ,
Cov
[R
,
(R
−
E(R
|R
))|
R
]
=
Cov
Y
,
(ξ
Y
)
i,2
M,2
M,2
1
1
i,2
(2)
σ2
]
[
[
]
1
2
⊥
2
=
Cov
R
,
(R
−
E(R
|R
))
R
=
Cov
Y
,
(ξ
Y
)
Y
.
i,2
M,2
M,2
1
1
i,2
(2)
(1)
σ2
b1i,M =
(15)
c1i,M
(16)
Moreover we need appropriate date 0 co-moments:
Definition 2 For asset i = 1, . . . , n, systematic date 0 co-moments are defined by
1
Cov [Ri,1 , (RM,1 − E(RM,1 ))] = Cov[Yi,1 , (ξ ⊥ Y(1) )],
2
σ
[
]
1
2
=
Cov
R
,
(R
−
E(R
))
= Cov[Yi,1 , (ξ ⊥ Y(1) )2 ],
i,1
M,1
M,1
2
σ
[
]
1
2
Cov
R
,
(R
−
E(R
|R
))
= Cov[Yi,1 , (ξ ⊥ Y(2) )2 ].
=
i,1
M,2
M,2
1
2
σ
b0i,M =
c0i,M
c0,∗
i,M
Note that similar to definition 1 for the one-period case, we also standardize the date 0 comoments by the scale of risk, see our discussion there. However, differently to definition 1 we
7
Here and throughout the paper all co-moments are expressed in terms of the random variables (Yi )1≤i≤n
and also in terms of original returns (Ri )1≤i≤n (appropriately rescaled by σ). Conditioning w.r.t. to the random
variable Y1 is the same as conditioning on R1 , since σ > 0.
16
only define the co-moments up to order 3, since we will study below only the first two terms
ai2 σ 2 and ai3 σ 3 in our expansion.
Note that c0,∗
i,M describes a form of intertemporal third order co-moment; it should not be
confused with the time-zero expectation of the market co-skewness at time 1, since
E[c1i,M ] =
[
]
1
2
Cov
R
,
(R
−
E(R
|R
))
= Cov[Yi,2 , (ξ ⊥ Y(2) )2 ].
i,2
M,2
M,2
1
2
σ
This term does not vanish if and only if the market return over the second period is conditionally
heteroskedastic given first period returns:
0,∗
ci,M
=
Aggregating gives
[
]]
1
⊥
Y(1) .
Cov
[R
,
V
ar[R
|R
]]
=
Cov
Y
,
V
ar[ξ
Y
i,1
M,2
1
i,1
(2)
2
σ
n
∑
0,∗
ξi ci,M
=
i=1
1
Cov [RM,1 , V ar[RM,2 |R1 ]] = LM .
σ3
(17)
(18)
Here, LM is the market leverage (or volatility feedback) effect. We expect it to be negative:
bad news on the market leads to soaring expectations of future volatility. The term c0,∗
i,M defined
in equation (17) describes the contribution of each asset to the market leverage effect, i.e. a
co-leverage (or volatility co-feedback) effect.
We solve for the date 0 risk premiums by plugging in the definition of W2 (σ), the value of risk
premium terms πi,1 (σ) implied by formulas (12), using θ(1) = ξ and solving first order conditions
(11) with respect to unknown premiums πi,0 (σ) when plugging in θ(0) = ξ; we are then able to
prove:
Theorem 2 The date 0 equilibrium risk premium πi,0 (σ) = E(Ri,1 ) − Rf =
acterized by
a0i2 =
1 0
b ,
τ0 i,M
a0i3 = −
ρ0 0
1 − ρ0 + γ0 0,∗
c +
ci,M
2 i,M
τ0
τ02
where
τ0
(
)
u′ qRf2
(
),
= −
qRf u′′ qRf2
(
)
(
)
u′ qRf2 u′′′ qRf2
(
)
ρ0 =
2[u′′ qRf2 ]2
and the preference term γ0 is defined in equation (14).
17
∑∞
j=2
a0ij σ j is char-
This theorem has interesting implications for equilibrium risk premiums8 . In this theorem 2
the preference parameters τ0 , ρ0 are analogous the preference parameters τ, ρ in theorem 1. The
main difference is that the terms in theorem 1 are defined using the (single-period) riskfree return
Rf , whereas those in theorem 2 make use of the two-period riskfree return Rf2 , since utility is
evaluated at wealth two periods from date 0.
Note that here we do not make explicit higher order terms of risk premium like a0i4 and a0i5 ;
this is different from theorem 1 for the one-period case. Although we consider only the first two
terms a0i2 , a0i3 , we can already see important differences in the two period situation compared to
the one period case of the previous subsection. The main message here is that dynamic pricing
introduces c0,∗
i,M , another concept of market co-skewness at date 0. This additional term is the
date 0 risk premium due to the sensitivity of changes in the date 1 risk premium through changes
in market leverage (volatility feedback). The price of volatility risk is given here through
1−ρ0 +γ0
.
τ02
0,∗
Note that the presence of ci,M
is typically due to intertemporal optimization behavior and
is characterized through the wealth tolerance ρ0 . (The impact of c0,∗
i,M on risk premiums is not
limited to wealth tolerance γ0 . For example, the pricing impact disappears in case of myopic
behavior implied by the logarithmic utility function, since then τ0 = Rf , ρ0 = 1, γ0 = 0.) We note
that a non-zero preference for positive skewness (ρ0 > 0) gives rise to risk premiums for asset i
not only for its market co-skewness but also for its market co-leverage. This matches the welldocumented empirical fact that leverage effect creates skewness through temporal aggregation
(see e.g. Meddahi and Renault (2004)).
3.3
Quadratic SDF
A convenient way to describe an asset pricing model is through the stochastic discount factor
(SDF), see e.g. Cochrane (2001), i.e. through a random variable H with the property that risk
premiums are given by
E[Ri ] − Rf = −Rf Cov(H, Ri ), for i = 1, . . . , n.
The literature usually considers time-series applications, where expectations are conditional
on the information available at each date. In line with our earlier analysis and for simplicity we
8
We only discuss the date 0 equilibrium risk premium and not the date 1 risk premium. The latter can be
interpreted through theorem 1 with a conditional viewpoint.
18
do not write out this dependence explicitly. Thus, essentially, our analysis appears to boil down
to a single-period analysis but the reader should keep in mind the conditional viewpoint. We
study first the SDF of the first subsection, i.e. the case of a single period case; then we study
the two-period extension of the second subsection and will see that it leads us to consider an
additional pricing factor in the SDF.
It is well known that the Sharpe-Lintner CAPM is tantamount to an SDF that is affine w.r.t.
the market return. To accommodate some widely documented departures from the CAPM,
empirical asset pricing may consider higher-order polynomials in the market return as a candidate
SDF. For instance a quadratic SDF should accommodate pricing of skewness, see, among others,
Harvey and Siddique (2000) and Dittmar (2002); up to a constant, a quadratic SDF, H(2), has
the form
2
Rf H(2) = λ̄1 RM + λ̄2 RM
,
(19)
for suitable parameters λ̄1 , λ̄2 . It is well-known that the quadratic SDF leads to a linear pricing
relationship for excess returns:
[
]
2
E [ri ] = λ1 Covt [ri , rM ] + λ2 Covt ri , rM
,
(20)
where ri = Ri −Rf , rM = RM −Rf stand for excess returns on asset i and the market, respectively.
It is tempting to apply the linear pricing relationship (20) to the squared market return; to
2
do so, let us assume for a moment and for purely illustrative reasons that rM
corresponds to a
2
2
portfolio available in the market. (We will discuss in the next section that RM
and thus rM
may
2
not be a portfolio in incomplete markets.) We denote by η the price and by r̄M = rM
/η − Rf
the excess return on the squared market portfolio; applying the linear pricing relationship (20)
to this excess return gives
(
)
(
)
[ 2 ]
2
2
rM
rM
2 rM
− Rf = λ1 Cov rM ,
+ λ2 Cov rM ,
.
E
η
η
η
The linear pricing relationship (20) can also be applied to the market excess return; this gives
a system of two equations that can be resolved:
2
2
2
V ar(rM
)E[rM ] − Cov(rM , rM
)(E[rM
] − ηRf )
,
2
2
V ar(rM )V ar(rM ) − (Cov(rM , rM ))2
2
2
V ar(rM )(E[rM
] − ηRf ) − Cov(rM , rM
)E[rM ]
=
.
2
2
V ar(rM )V ar(rM ) − (Cov(rM , rM ))2
λ1 =
λ2
19
One may be tempted to set η = 0; this characterization of λ1 , λ2 then coincides with formulas
(7b, 7c) put forward by Harvey and Siddique (2000). However, this appears to be at odds with
a no-arbitrage condition, since η is the price of a strictly positive payoff and should therefore
be strictly positive. Also, when markets are incomplete we will stress in the next section that
the squared market returns may not be in the asset span, such that its price cannot be observed
easily.
Our analysis in subsection 3.1 provides a foundation for the common practice of quadratic
SDF and allows us to relate the terms λ1 , λ2 to preference parameters. Based on our analysis of
ai2 σ 2 + ai3 σ 3 we can identify the parameters λ̄1 , λ̄2 in the quadratic SDF of equation (19); we
have, up to terms higher then σ 3 :
1
ρ
bi,M σ 2 − 2 ci,M σ 3 = ai2 σ 2 + ai3 σ 3 = E[Ri ] − Rf = −Rf Cov(H(2), Ri ).
τ
τ
Based on definition 1, this puts forward the quadratic SDF up to a constant
1
ρ
Rf H(2) = − RM + 2 (RM − E(RM ))2 .
τ
τ
(21)
Our characterization of the SDF in equation (21) allows us to identify to the SDF in equation
(19), up to a constant:
1
ρ
ρ
λ̄1 = − − 2 2 E [rM ] , λ̄2 = 2 .
τ
τ
τ
The price of skewness is contained in the sensitivity λ̄2 : when it is zero we are back in a meanvariance pricing world, while λ̄2 ̸= 0 leads to mean-variance-skewness pricing. Note that here λ̄2
is something like a structural invariant, only varying through the value of preference parameters
computed from the derivatives of the utility function at Rf . Ultimately, estimating the size of
λ̄2 is at the core of empirical studies, see, e.g., Harvey and Siddique (2000).
Overall, we find that a single period analysis of a representative agent confirms the common
practice of quadratic SDF. Ultimately, we will elaborate below from two angles that this paradigm
breaks down: first we elaborate in the remainder of this (sub-)section that this may not hold in
a two period analysis of a representative agent; second, section 5 will take up the issue in great
detail with heterogeneous agents in a single period analysis and point out that there are priced
factors in incomplete markets that do not show up in complete markets.
Let us now study the implications of our two-period extension in the previous subsection;
for this we denote the SDF by H̃(2) to distinguish it from the prior SDF. We derive from the
20
analysis of the first two terms ai2 σ 2 + ai3 σ 3 in theorem 2 that the date 0 one-period risk premium
is (up to higher order terms)
(
)
1
ρ0
1 − ρ0 + γ0 0,∗ 3
−Rf Cov H̃(2), Ri,1 = E (Ri,1 ) − Rf = b0i,M σ 2 − 2 c0i,M σ 3 +
ci,M σ .
τ0
τ0
τ02
Up to a constant, this puts forward the date 0 one-period SDF
H̃(2) = H̃(2, 1) + H̃(2, 2),
where
Rf H̃(2, 1) = Rf H(2) = −
Rf H̃(2, 2) = −
1
ρ0
RM,1 + 2 (RM,1 − ERM,1 )2 ,
τ0
τ0
1 − ρ0 + γ0
V ar [RM,2 |R1 ] .
τ02
Compared to the one-period SDF in the static model, there is an additional pricing factor
which is the volatility risk factor V ar [RM,2 |R1 ]. The price of risk of this factor is determined
not only by the risk preferences (risk and skew tolerance), but also through a wealth effect via
the wealth tolerance γ0 .
One message of this section is that a two period analysis may be at odds with the common
quadratic SDF assumption, except if one maintains two rather restrictive assumptions about
the volatility risk factor V ar[RM 2 |R1 ]. These assumptions are statistical in nature and not
underpinned by any economic theory: First, we need to assume that an aggregated volatility
model is optimal:
V ar[RM 2 |R1 ] = V ar[RM 2 |RM 1 ]
(22)
Second, even under this assumption we only get the quadratic SDF specification by assuming an
ARCH(1) specification (Engle (1982)) for this aggregated volatility model:
2
V ar[RM 2 |RM 1 ] = ϖ + ςRM
1
Even an asymmetric ARCH (Glosten et al. (1993)), where the sign of RM 1 would matter (to accomodate an aggregate leverage effect) would invalid the naive quadratic SDF. Note in addition
that the aggregation assumption (22) is all about the cross section of risky assets. We always assume that investors are homogeneous regarding their expectation of future returns. Aggregation
of heterogeneous beliefs and/or asymmetric information is beyond the scope of this paper.
21
4
Portfolio separation and pricing of skewness risk with
heterogeneous agents
This previous section studied the case of a single (representative) investor. Throughout the
remainder of this paper we assume the economy is populated by S (heterogeneous) investors
s = 1, ..., S; each investor s maximizes expected utility E[us (Ws )] of her terminal wealth Ws
obtained by investing her initial wealth qs in a portfolio.
This section discusses demand of our heterogeneous investors and the pricing of skewness
risk when we focus on the first two terms ai2 σ 2 + ai3 σ 3 ; along the way we also look into the
impact of market incompleteness. We study this in two separate subsections first for the case
of heterogeneous investors over one period and then for the case of heterogeneous investors over
two periods.
4.1
Case of heterogeneous investors over one period
Throughout, this subsection we assume that there is only one period of investment. The portfolio
of each investor s = 1, ..., S is defined in terms of shares of wealth invested θs = (θis )1≤i≤n ; this
gives terminal wealth
[
Ws = qs Rf +
n
∑
]
θis (Ri − Rf ) .
(23)
i=1
Note that this is the heterogeneous investor analogue (for each investor s = 1, ..., S) of the single
period wealth dynamic that we considered in the representative investor case, equation (5). The
demand θs of investor s for risky assets solves the first order conditions of her expected utility
maximization, where the first order condition for each investor s = 1, ..., S is merely equation
(2) written out for the utility function us and terminal wealth Ws based on demand θs of each
investor.
Samuelson (1970) characterized the impact of higher order return moments on the demand
for risky assets through a series expansion of the optimal portfolio θs = (θis )1≤i≤n :
θis =
∞
∑
θisj σ j .
(24)
j=0
In section 2 we assumed that the total offer of risky assets was a fixed vector ξ = (ξi )1≤i≤n
∑
of shares of the total amount S q̄ = Ss=1 qs of invested wealth and we continue to adopt this
22
assumption. Note that
∑S
s=1 θis qs
is the aggregate demand of any asset i = 1, ..., n. Therefore,
we have the following market clearing conditions for each risky asset i = 1, ..., n and all orders
j > 0:
S
∑
θis0 qs = ξi S q̄,
and
s=1
S
∑
θisj qs = 0.
(25)
s=1
This differs from our representative investor analysis (the previous section) in several ways.
There, the total offer was ξ which effectively limited us to θ = ξ, i.e. all terms of order higher
than 1 had to vanish. Here, this does not have to be the case at the level of individual investors;
this means that investors can use higher order terms to hedge risks stemming from powers of
market return not being in the asset span (in incomplete markets); this in turn will have an
impact on risk premiums that only shows up when investors are heterogeneous.
The market clearing conditions (25) together with the first order conditions for each investor
determine the equilibrium asset demand and pricing of financial assets; the coefficients aij , j ≥ 2
of the risk premium expansion (3) solve for all levels σ of risk:
E [u′s (Ws )(Ri − Rf )] = 0, for i = 1, ..., n,
[
]
n
∑
subject to Ws (σ) = qs Rf +
θis (σ) · (Ri (σ) − Rf ) ,
(26)
i=1
where Ri (σ) − Rf = πi (σ) + σYi , for i = 1, ..., n.
Samuelson’s key insight that we extend in this paper is the following: for any investor s,
the first K coefficients θisj , j = 0, ..., K − 1, only depend on the first (K + 1) derivatives of the
utility functions us , s = 1, ..., S, computed at the level of wealth qs Rf . (This level can be seen
as the terminal wealth resulting from investing the initial wealth qs only in the safe asset.) This
permits successive determination of demand and market clearing, starting from terms of order
zero and going up one order at a time. We are then able to prove:
Theorem 3 The equilibrium risk premium πi (σ) = E(Ri ) − Rf =
ai2 =
∑∞
j=2
aij σ j fulfills
1
ρ̄
bi,M , ai3 = − 2 ci,M ,
τ̄
τ̄
(27)
where individual (τs ) and average (τ̄ ) risk tolerance coefficients as well as individual (ρs ) and
average (ρ̄) skew tolerance coefficients are defined by
∑S
S
qs τs ρs
1 ∑
u′s (qs Rf )
u′s (qs Rf )u′′′
s (qs Rf )
, τ̄ =
.
τs = − ′′
, ρ̄ = ∑s=1
τ s qs , ρ s =
2
S
qs us (qs Rf )
S q̄ s=1
2 (u′′s (qs Rf ))
s=1 qs τs
23
(28)
The equilibrium demand of investor s is given through
θs0 =
τs
ξ,
τ̄
θs1 =
τs (ρs − ρ̄) −1
Σ c,
τ̄ 2
(29)
where θs0 = (θis0 )1≤i≤n , θs1 = (θis1 )1≤i≤n describe the zero, first order demand vector and c =
(ci,M )1≤i≤n describes the vector of market co-skewness.
Theorem 3 has interesting implications for equilibrium risk premiums. Approximating the
risk premium (E(Ri ) − Rf ) by its first two terms σ 2 ai2 + σ 3 ai3 is similar to extending the CAPM
to an asset pricing model that takes into account skewness risk. Skewness risk is priced in
equilibrium as soon as the properly defined average skew tolerance is non-zero and some assets
display a non-zero co-skewness (while the market return may well be symmetric). Our result
displays the aggregate impact on the skewness premium in case of investors heterogeneity.
Heterogeneity across investors’ risk tolerances and skew tolerances τs , ρs (s = 1, ..., S) may
come either from heterogeneity of their utility functions us or from heterogeneity of their invested wealth qs . In both cases we find here the well-known result that heterogeneity does not
really matter for mean-variance-skewness pricing; only the average risk tolerance coefficient τ̄ , ρ̄
matters. Note in particular that for homogeneous investors (same utility function u = us , same
wealth q = q̄ invested), the risk premium on variance and on skewness risk would simply be that
of Theorem 1.
Let us now take a look the implications of theorem 3 for equilibrium asset demand. We
have seen that the Sharpe-Lintner CAPM gives a correct picture of actual risk premiums insofar
as we focus on the first term ai2 σ 2 of their series expansion. Theorem 3 shows that the same
applies to the equilibrium demand for risky assets, i.e. a standard mean-variance approach is
sufficient insofar as we focus on the first term θs0 of the series expansion for the equilibrium
demand of investor s. This is the celebrated two-fund separation theorem, where all investors
hold the same portfolio of risky assets defined by portfolio weights ξ. By analogy with the CAPM
relationship, let us study an additional fund defined by portfolio weights ξ sk that lead to a return
∑
Rsk = Rf + ni=1 ξisk (Ri − Rf ) with the property
σ 3 aj3 = −
ρ̄
Cov(Rj , Rsk ), for all j = 1, . . . , n.
τ̄ 2
24
(30)
Note that this equation defines ξ sk as a unique solution of a linear system of n equations with
non-singular matrix V ar(R). Theorem 3 states that
ξ sk = σΣ−1 c.
This new portfolio plays the same role for pricing skewness risk that the market portfolio played
to price (co-)variance risk. Therefore, throughout this paper we refer to it as the skewness
portfolio. Not surprisingly, it also appears in investors’ equilibrium demand for risky assets, i.e.
we find:
σθs1 =
τs (ρs − ρ̄) sk
ξ .
τ̄ 2
(31)
Equations (29) establish a three-fund theorem: All investors hold the same two portfolios
of risky assets (so-called mutual funds) defined by portfolio weights ξ and ξ sk , respectively.
The only difference in actual holdings comes from wealth heterogeneity (ceteris paribus, the
amount invested is proportional to wealth), heterogeneity in risk tolerances (the amount invested
is proportional to risk tolerance at this wealth level) and, regarding the second mutual fund,
heterogeneity in skew tolerances (the amount invested is proportional to relative spread between
the skew tolerance of investor s and the average one). Thus, an investor with stronger or weaker
preferences for skewness than the average must underdiversify with respect to mean-variance
efficiency.
Recall that the wealth weighted sum of τs ρs in relation to the wealth weighted sum of τs gives
the average skew tolerance coefficient ρ̄. Therefore, although investors’ holdings in the skewness
portfolio may be different from zero, aggregate demand of it vanishes9 .
It is worth elaborating in more detail why investors will have non-zero positions in the skewness portfolio when their preferences for skewness depart from the average skewness preference.
Comparing (27) and (30) shows that for all assets i = 1, ..., n:
[
]
Cov[Ri , Rsk ] = Cov Ri , (RM − E(RM ))2 .
(32)
In other words, we have the following decomposition for the squared market return:
(RM − E(RM ))2 = Rsk − E(Rsk ) + T sk ,
9
(33)
Essentially this is due to our simplifying assumption that the offer of risky assets does not depend on the
scale parameter σ. Changing this assumption may introduce higher order dependence.
25
where the so-called tracking error (T sk − E(T sk )) is uncorrelated with all traded asset returns
Ri , i = 1, ..., n. Up to an additive constant, the skewness portfolio return Rsk can therefore be
interpreted as an affine regression of the squared (demeaned) market return on the traded asset
returns. In other words, the skewness portfolio is the mutual fund that is best able to track the
squared market return; thus, demand for the skewness portfolio is induced by the demand for
tracking the squared market return10 .
In complete markets, the squared market return can be replicated through traded assets and
the tracking error disappears. In incomplete markets, however, the squared market return may
not be in the asset span. The variance V ar(T sk ) of the tracking error can then be interpreted
as a measure of market incompleteness. The bigger this variance, the less accurate is the feasible hedge of the squared market return that a representative agent would like to hold for the
sake of skewness preferences (see e.g. Dittmar (2002)). This interpretation in terms of market
incompleteness will be confirmed in the next section.
4.2
Case of heterogeneous investors over two periods
This subsection extends the discussion of equilibrium with heterogeneous investors from one to
two periods. Our extension here is analogous to the extension in the representative agent case
from one to two periods in subsection 3.2.
This subsection continues to consider s = 1, . . . , S investors who maximize their expected
utility E[us (Ws )] derived from the terminal wealth Ws that they obtain by investing their initial
wealth qs in a portfolio. The difference to the previous subsection is that we take into account that
this terminal wealth may be the result of a dynamic investment strategy over several consecutive
periods. For sake of expositional simplicity, we consider only two periods with constant risk-free
rate Rf = Rf,0 = Rf,1 : the first period runs from date 0 to date 1, the second period with runs
from date 1 to date 2. In other words, the dynamic portfolio of the representative investor is
1
0
)1≤i≤n to give terminal wealth:
, θis
defined in terms of shares of wealth invested θs = (θis
[
]
[
]
n
n
∑
∑
1
0
Ws,2 = Ws,1 · Rf +
(Ri,2 − Rf ) , Ws,1 = qs · Rf +
(Ri,1 − Rf ) . (34)
θis
θis
i=1
i=1
10
In line with this reasoning, a careful examination of option markets would show that preferences for skewness
introduce non-zero demand in asymmetric assets like options, see Judd and Leisen (2010).
26
The second period portfolio holdings may depend on the realization of returns over the first
period, but for simplicity of exposition we do not write out this dependence explicitly throughout
this subsection.
We continue to use the same scale-location model for the vector of returns that we defined
recursively over consecutive periods in equations (8, 9) of subsection 3.2, i.e.
R(1) = (Ri,1 )1≤i≤n = E(R(1) ) + σY(1) , with E(Y(1) ) = 0,
(
)
(
)
and R(2) = (Ri,2 )1≤i≤n = E R(2) Y(1) + σY(2) , with E Y(2) Y(1) = 0.
(35)
(36)
As in subsection 3.2 we adopt the simplifying assumption that ξ 1 = ξ, see our discussion there.
The associated market returns RM,1 and RM,2 over both periods are then given as
RM,1 = ξ ⊥ E(R(1) ) + σξ ⊥ Y(1) ,
(
)
RM,2 = ξ ⊥ E R(2) Y(1) + σξ ⊥ Y(2) .
Similar to the series expansion of demand in the equation (24) of the previous subsection
we characterize the impact of these higher order moments on the demand for risky assets at
0
both dates 0 and 1 through a series expansion of the optimal portfolio θs0 = (θis
)1≤i≤n and
1
θs1 = (θis
)1≤i≤n , respectively:
0
θis
=
∞
∑
0
θisj
σj
and
1
θis
j=0
=
∞
∑
1
θisj
σj .
(37)
j=0
Aggregate demand of any asset i = 1, ..., n is
∑S
0
s=1 θis qs
and
∑S
1
s=1 θis Ws,1
at dates 0 and 1.
Therefore, we have the following market clearing conditions for each risky asset i = 1, ..., n and
all orders j > 0:
S
∑
0
θis0
qs
= ξi S q̄,
and
s=1
S
∑
1
θis0
Ws,1
= ξi
s=1
S
∑
Ws,1 ,
and
s=1
S
∑
s=1
S
∑
0
θisj
qs = 0
at date 0;
(38)
1
θisj
Ws,1 = 0 at date 1.
(39)
s=1
Based on Lemma 1, the scale parameter σ allows us to describe the pattern of risk premiums
from their series expansions:
(
)
πi,1 (σ) = E Ri,2 | Y(1) − Rf =
∞
∑
a1ij σ j
and πi,0 (σ) = E (Ri,1 ) − Rf =
∞
∑
j=2
j=2
27
a0ij σ j .
(40)
At date 0, every investor s = 1, . . . , S faces the following dynamic optimization program:
E[Js (Ws,1 (σ))],
max
E[us (Ws,2 (σ))] = max
θs0
θs0 ,θs1
]
where Js (Ws,1 (σ)) = max
E[u(W
s,2 (σ)) Y(1) ,
1
θs
subject to the associated wealth dynamics in equation (34). Using the date 1 indirect utility
function Js defined above and the law of iterated expectations, we can then write the investor’s
first order conditions for date 0 portfolio optimization for i = 1, . . . , n as
[
]
[
]
∂Ws,2
∂Js ∂Ws,1
′
E us (Ws,2 (σ))
(πi,0 (σ) + σYi,1 ) = E
= 0,
0
∂Ws,1
∂Ws,1 ∂θis
(41)
where the terminal wealth level Ws,2 = Ws,2 (σ) is induced by the portfolio choice according to
(34). This describes the date 0 first order conditions for each investor s = 1, . . . , S. These date 0
first order conditions, together with the date 0 market clearing conditions (38) determine jointly
the date 0 equilibrium allocation (demand, risk premiums)11 .
We are solving this through backward induction. Our two date return setup here mirrors the
two date return setup with representative agent in subsection 3.2. Therefore, we continue to use
the definitions of date 1 market beta and market co-skewness that we introduced in equations
(15, 16) analogously definition 1. At date 1, given the observed realization of the state vector
Y(1) , each investor determines her demand based on the return prospects whose randomness is
defined by the scale parameter σ and the conditional probability distribution of Y(2) given Y(1) ;
the market clearing condition (38) then determines the equilibrium risk premiums, which also
determines the optimal demand of each investor s = 1, . . . , S.
To apply the one-period market results derived in the former subsection we define first individual and aggregate preference parameters through the date 1 (aggregate) risk tolerance and
(aggregate) skew tolerance functions
u′ (wRf ) u[3] (wRf )
u′ (wRf )
, ρs,1 (w) =
τs,1 (w) = − ′′
wu (wRf )
2[u′′ (wRf )]2
∑S
∑S
ws ρs,1 (ws )τs,1 (ws )
s ws τs,1 (ws )
, ρ̄1 (w1 , . . . , wS ) = s ∑S
τ̄1 (w1 , . . . , wS ) =
∑S
s ws
s ws τs,1 (ws )
(42)
(43)
Note that these are analogous to the risk and skew tolerances that we used in the single period case, see equation (28) in theorem 3. The individual risk and skew tolerance functions
11
The date 1 equilibrium is defined analogous the equilibrium in the previous subsection, but with a conditional
viewpoint. Our focus is on the date 0 equilibrium and therefore we do not write this out explicitly here.
28
will be evaluated at Ws,1 ; the aggregate risk and skew tolerance functions will be evaluated at
(Ws,1 )s=1,...,S .
We conclude from the one-period market results derived in the former subsection that the
(conditional) date 1 equilibrium risk premium for each asset i = 1, ..., n in the characterization
of equation (40) is
a1i2 =
1
b1 ,
τ̄1 (W1,1 , . . . , WS,1 ) i,M
a1i3 = −
ρ̄1 (W1,1 , . . . , WS,1 ) 1
c .
τ̄12 (W1,1 , . . . , WS,1 ) i,M
(44)
We stress that τ̄1 , ρ̄1 are random variables conditional on the realization of wealth (Ws,1 )s=1,...,S
for all investors. In the two-period case of the representative agent (subsection 3.2) a wealth effect
appeared that we characterized through a preference parameter called (date 0) wealth tolerance.
Here we define analogously for each investor s = 1, . . . , S the date 0 wealth tolerance γs by
setting
γs,0
∑S
γs qs τs,0
τs,0 ∂ τ̄1
= qs
(qs Rf ), γ̄0 = ∑s=1
.
S
τ̄0 ∂ws
s=1 qs τs,0
(45)
Here we used the date 0 risk tolerances
τs,0
(
)
∑S
u′s qs Rf2
s=1 qs τs,0
(
) , τ̄0 = ∑
= −
.
S
2
′′
Rf qs us qs Rf
s=1 qs
In addition, we introduce the date 0 skew tolerances
(
)
(
)
∑S
2
u′s qs Rf2 u′′′
q
R
s
s
f
s=1 qs ρs,0 τs,0
ρs,0 =
, ρ̄0 = ∑
[ (
)]
2
S
2 u′′s qs Rf2
s=1 qs τs
(46)
(47)
Our two date return setup here mirrors the two date return setup with representative agent
in subsection 3.2. Therefore, we continue to use the definitions of date 0 systematic co-moments
introduced in definition 2; for completeness we repeat these here:
1
Cov [Ri,1 , (RM,1 − E(RM,1 ))] = Cov[Yi,1 , (ξ ⊥ Y(1) )],
σ2
[
]
1
2
=
Cov
R
,
(R
−
E(R
))
= Cov[Yi,1 , (ξ ⊥ Y(1) )2 ],
i,1
M,1
M,1
σ2
[
]
1
2
=
Cov
R
,
(R
−
E(R
|R
))
= Cov[Yi,1 , (ξ ⊥ Y(2) )2 ].
i,1
M,2
M,2
1
σ2
b0i,M =
c0i,M
c0,∗
i,M
We are then able to prove:
29
Theorem 4 The date 0 equilibrium risk premium πi,0 (σ) = E(Ri,1 ) − Rf =
∑∞
j=2
a0ij σ j is char-
acterized by
a0i2 =
1 0
b ,
τ̄0 i,M
a0i3 =
1 − ρ̄0 + γ̄0 0,∗ ρ̄0 0
cM − 2 cM .
τ̄02
τ̄0
(48)
where τs,0 , τ̄0 , ρs,0 , ρ̄0 , γs,0 , γ̄0 are given through equations (45-47). The date 0 equilibrium demand
of investor s = 1, . . . , S fulfills
τs,0
ξ,
τ̄0
}
{
τs,0 (ρs,0 − ρ̄0 ) −1 0
τs,0 (ρs,0 − ρ̄0 ) τs,0 (γs,0 − γ̄0 )
0,∗
Σ−1
=
Σ0 cM +
−
0 cM ,
2
2
2
τ̄0
τ̄0
τ̄0
0
=
θs0
(49)
0
θs1
(50)
0
0
where θs0
= (θis0 )1≤i≤n and θs1
= (θis1 )1≤i≤n describe the (date 0) zero and first order demand
vector and Σ0 the (date 0) variance matrix V ar(R(1) ).
This theorem has interesting implications for equilibrium risk premiums12 . Although investors
may be heterogeneous in their preference parameters τs , ρs , γs (s = 1, . . . , S), we find that heterogeneity does not really matter for pricing; only aggregate preference parameters τ̄0 , ρ̄0 , γ̄0 are
relevant. In addition, we note that with these aggregate preference parameters, theorem 4 (two
period case, heterogeneous investors) matches the pricing result of theorem 2 (two-period case,
representative investor). In line with this, but looking at it from a different angle we compare
theorem 4 with theorem 3 and find that dynamic pricing introduces c0,∗
M . In subsection 3.2 we
argued that this concept of market co-skewness is related to market leverage (volatility feedback
effect) and referred to c0,∗
M as the co-leverage (or volatility co-feedback) effect.
Let us now take a look at the implications of theorem 4 for equilibrium asset demand. We
discussed in the previous subsection after theorem 3 that skewness leads to a three-fund theorem.
Analogous to there, we define a fund with portfolio weights ξ0sk leading to a return R1sk =
∑
Rf + ni=1 ξisk (Ri,1 − Rf ) with the property that the contribution of the following part to the
risk premium σ 3 a0i3 is given through:
−
ρ̄0 0 3
ρ̄0
0
c σ = − 2 Cov(Ri,1 , R1sk ) for i = 1, . . . , n , i.e. ξ0sk = σΣ−1
0 cM ;
2 i,M
τ̄0
τ̄0
12
(51)
We only discuss the date 0 equilibrium demand and risk premiums and not the date 1 allocation. The latter
can be interpreted through theorem 3 with a conditional viewpoint.
30
we refer to ξ0sk as the date 0 skewness portfolio. We note that Cov(Ri,1 , Rsk ) = Cov(Ri,1 , (RM,1 −
E(RM,1 ))2 ), such that there is a decomposition of the squared market return into the skewness
portfolio with return R1sk and a tracking error T1sk − E(T1sk ) that is uncorrelated with all traded
asset returns Ri , i = 1, . . .. The skewness portfolio can therefore be seen as the mutual fund
best able to track the squared market return. We find, again, that investors hold at date 0
the same portfolio of risky assets (so-called mutual funds) defined by portfolio weights ξ and
ξ0sk , respectively. As before, the difference in holdings comes from heterogeneity in risk and
heterogeneity in skew tolerances.
However, in our dynamic extension, an additional term due to the co-leverage effect shows
up in demand. Similarly to the skewness portfolio we define a fund with portfolio weights ξ0L
∑
leading to a return R1L = Rf + ni=1 ξiL (Ri,1 − Rf ) with the property that the contribution of the
following part to the risk premium σ 3 a0i3 is given through:
1 − ρ̄0 + γ̄0 0,∗ 3 1 − ρ̄0 + γ̄0
0,∗
ci,M σ =
Cov(Ri,1 , R1L ) for i = 1, . . . , n , i.e. ξ0L = σΣ−1
0 cM ;
τ̄02
τ̄02
(52)
we refer to ξ0L as the date 0 leverage portfolio. We note that
Cov(Ri,1 , RL ) = Cov(Ri,1 , (RM,2 − E(RM,2 |R1 ))2 ) = Cov(Ri,1 , V ar(RM,2 |R1 )) = σ 3 c0,∗
i,M ,
such that aggregating across the market gives
n
∑
ξi Cov(Ri,1 , RL ) = Cov [RM,1 , V ar[RM,2 |R1 ]] = LM ,
i=1
i.e. the market leverage (volatility feedback) effect, see also equation (18). Therefore, we view this
as a decomposition of the market leverage (volatility feedback) V ar(RM,2 |R1 ) into the leverage
portfolio with return R1L and a tracking error T1L − E(T1L ) that is uncorrelated with all traded
asset returns Ri , i = 1, . . . , n. The leverage portfolio can therefore be seen as the mutual fund
best able to track the market leverage (volatility feedback) V ar(RM,2 |R1 ).
Ultimately, theorem 4 provides a four-fund theorem: all investors hold at date 0 the same
portfolio of risk assets (so-called mutual funds) defined by portfolio weights ξ, ξ0sk and ξ0L , respectively; the difference in holdings comes from heterogeneity in risk, heterogeneity in skew
tolerances and heterogeneity in wealth tolerances. Our dynamic extension of the theorem 4
leads us consider one more (mutual) fund: the leverage portfolio. The leverage portfolio can be
31
seen as providing some hedge against the market leverage (volatility feedback), i.e. it protects
against future changes in market volatility. This is a form of intertemporal hedge demand that
is common in dynamic portfolio selection problems, see, e.g. Liu (2007), and Chacko and Viceira
(2005).
5
Unspanned SDF in polynomials of market return
Our analysis showed that a dynamic extension introduces an additional priced factor that is
overlooked in the standard approach of a Taylor series expansion of the representative agents
utility function in single period models. This section focuses on the single-period setup and
studies whether tracking errors in incomplete markets lead to additional priced factors that have
been overlooked in the current literature.
So far, we approximated the risk premium in the single period case through the first two terms
ai2 σ 2 +σai3 σ 3 of its expansion (3); in this approximation, a polynomial of degree two in the market
return is a sufficient statistic to define the SDF, see equation (21); this is in accordance with the
representative agent model of Dittmar (2002). However, this ignores the impact of higher order
terms in the expansion (3). This section carries out a careful investigation of higher order terms
aij σ j (j = 4, 5) in the expansion (3) and shows that polynomials in the market return do not
capture completely the pricing impact of investors’ heterogeneity in incomplete markets.
5.1
The spanning issue with polynomials of market return
Following the logic of Harvey and Siddique (2000) as well as based on the representative agent
models in Dittmar (2002) and Chung et al. (2006), we may expect that additional terms in
the expansion (3) lead to a characterization of the risk premium on asset i, i = 1, . . . , n as a
linear function that consists not only of the standard market beta σ 2 bi,M and of the standard
co-skewness σ 3 ci,M , but also of the co-kurtosis σ 4 di,M , the co-pentosis σ 5 fi,M , and so on, as we
go further in the expansion. This would lead to SDFs represented as linear functions of RM and
(RM − E(RM ))j , j = 2, 3, 4, ..., i.e. as polynomials in the market return RM .
The problem with this reasoning is that it overlooks a form of market incompleteness, namely
the impossibility to get a perfect hedge for powers of the market return from the payoff of linear
portfolios. Let us characterize this issue for the power j = 2. As explained in section 4 above
32
(see equation (33)) there is in general a tracking error:
(RM − E(RM ))2 = Rsk − E(Rsk ) + T sk .
By construction, the tracking error (T sk − E(T sk )) is uncorrelated with any primitive asset
return, so that the squared market return captures perfectly the price of skewness risk:
σ 3 ci,M = Cov[Ri , Rsk − E(Rsk )] = Cov[Ri , Rsk ], for all i = 1, ..., n.
Unfortunately, this may not be true anymore when it comes to the price of kurtosis risk and
that of pentosis risk. In the former case we have
σ 4 di,M = Cov[Ri , (RM − E(RM )(Rsk − E(Rsk )] + Cov[Ri , (RM − E(RM ))T sk ],
while we find in the latter case
σ 5 fi,M = Cov[Ri , (Rsk − E(Rsk ))2 ] + Cov[Ri , (T sk )2 ] + 2Cov[Ri , (Rsk − E(Rsk ))T sk ]
= −Cov[Ri , (Rsk − E(Rsk ))2 ]
+2Cov[Ri , (RM − E(RM ))2 (Rsk − E(Rsk ))] + Cov[Ri , (T sk )2 ].
This is the reason why we may expect that proper pricing of higher order risks will involve
pricing factors like (RM − E(RM ))(Rsk − E(Rsk ), (Rsk − E(Rsk ))2 and (RM − E(RM ))2 (Rsk −
E(Rsk )) in order to span the otherwise unhedgeable risk brought by (RM −E(RM ))T sk and (T sk )2
respectively. In other words, in case of market incompleteness, spanning the SDF may require
not just polynomials in the single variable (RM − E(RM )), but polynomials in two variables:
(RM − E(RM )) and, in addition, (Rsk − E(Rsk )). Note that only these two variables show up
because we have limited our analysis so far to the tracking error on the skewness portfolio.
Higher order terms in the expansions take polynomials with additional variables corresponding to returns on additional mimicking portfolios. These additional mimicking portfolios consist
of two types: first, there are mimicking portfolios of higher powers of returns. We will illustrate
this remark in subsection 5.3 below by introducing, similarly to the skewness portfolio, a kurtosis
portfolio with return Rkurt and a tracking error (T kurt − E(T kurt )) on it:
(RM − E(RM ))3 = Rkurt − E(Rkurt ) + T kurt ,
33
(53)
where T kurt is uncorrelated with all primitive asset returns Ri , i = 1, ..., n. Second, there are
mimicking portfolios for products of market returns with tracking errors. In the next subsection
we will introduce, similarly to the skewness portfolio, a co-cross skewness portfolio portfolio with
return R∗sk and a tracking error (T ∗sk − E(T ∗sk )) on it:
(RM − E(RM ))(Rsk − E(Rsk )) = R∗sk − E(R∗sk ) + T ∗sk ,
(54)
where T ∗sk is uncorrelated with all primitive asset returns Ri , i = 1, ..., n. We will then illustrate
in subsection 5.3 the pricing impact of products of the market return with this co-cross skewness
portfolio. Overall, we expect pricing kernels that capture pentosis risk to be polynomials of four
variables: (RM − E(RM )), (Rsk − E(Rsk )), R∗sk − E(R∗sk ) and (Rkurt − E(Rkurt )).
Of course, market incompleteness, as manifest in the non-zero tracking errors (T sk − E(T sk )),
(T ∗sk −E(T ∗sk )) and (T kurt −E(T kurt )) will have a pricing impact only if it matters for individual
investors in equilibrium, i.e. these investors are heterogeneous in terms of preferences for higher
order moments.
5.2
Co-kurtosis pricing and unspanned co-cross skewness
In this subsection, we go one step further than subsection 3.3. We want to characterize the pricing
factors which matter to describe the approximation of the risk premium E(Ri ) − Rf through
the first three terms ai2 σ 2 + ai3 σ 3 + ai4 σ 4 of its expansion (3). We show that the addition of
the third term σ 4 ai4 may introduce two additional pricing factors: First, we find an additional
power of the market return, namely the cubic one, to go beyond quadratic SDF of section 4, as
in the representative agent model of Dittmar (2002). However, second, we also find the co-cross
skewness factor (RM − E(RM ))(Rsk − E(Rsk )). This factor will be priced in equilibrium if and
only if investors display some heterogeneity in their skew tolerances.
Here we extend definitions of aggregate characteristics of preferences introduced earlier as
follows:
∑S
∑S
ρ̄ =
qs τs ρs
,
∑s=1
S
q
τ
s
s
s=1
κ̄ =
qs τs κs
,
∑s=1
S
q
τ
s
s
s=1
∑S
V ar(ρ) =
s=1 qs τs (ρs −
∑S
s=1 qs τs
ρ̄)2
.
(55)
We remind the definition of individual skew-tolerance ρs and define kurtosis-tolerance κs by
ρs =
u′s (qs Rf )u′′′
s (qs Rf )
,
′′
2[us (qs Rf )]2
κs =
[u′s (qs Rf )]2 u′′′′
s (qs Rf )
.
′′
6[us (qs Rf )]3
The approximation E(Ri ) − Rf ≈ ai2 σ 2 + ai3 σ 3 + ai4 σ 4 leads to complete theorem 3 by
34
(56)
Theorem 5 With heterogeneous investors s = 1, . . . , S we complete the characterization of the
∑
j
equilibrium risk premium πi (σ) = E(Ri ) − Rf = ∞
j=2 aij σ in theorem 3 by
σ 4 ai4 =
κ̄
1 − 3ρ̄
V ar(ρ)
di,M +
VM bi,M − 2
Cov[Ri , (RM − E(RM ))(Rsk − E(Rsk )], (57)
3
3
τ̄
τ̄
τ̄ 3
where τ̄ is defined in equation (28) and ρ̄, κ̄, V ar(ρ) are defined in equation (55).
Comparing the description of ai4 in theorem 5 to that in theorem 1, i.e. with the fourth
order risk premium term in the case of a representative agent, we see that the covariance of
individual asset returns i with the co-cross skewness factor (RM − E(RM ))(Rsk − E(Rsk )) enters
into pricing. We will discuss below that this covariance may not vanish. Before doing so, we
take a look at risk premiums from the perspective of SDF and priced factors: the approximation
(57) puts forward an SDF H(3) such that up to constants:
H(3) ≈ H(3, 1) + H(3, 2), where
(58)
{
}
1 1 − 3ρ̄
ρ̄
κ̄
(−Rf )H(3, 1) =
+
V ar(RM ) RM − 2 (RM − E(RM ))2 + 3 (RM − E(RM ))3 ,
3
τ̄
τ̄
τ̄
τ̄
V ar(ρ)
(−Rf )H(3, 2) = −2
(RM − E(RM ))(Rsk − E(Rsk )).
(59)
τ̄ 3
Here, H(3, 1) is the classical cubic SDF provided by representative agent models, see, e.g.
Dittmar (2002). As usual, the three pricing factors RM , (RM −E(RM ))2 and (RM −E(RM ))3 have
prices proportional to risk aversion, skew tolerance and kurtosis tolerance, respectively. Note that
adding the expansion term σ 4 a4 has led us to adding the correction term (1 − 3ρ̄)V ar(RM )/τ̄ 3
to the standard weight (1/τ̄ ) of the market return; note also, however, that this correction term
is of lower order in small noise expansions since V ar(RM ) is proportional to σ 2 .
More important, by comparison with the classical SDF that is a polynomial in the market
return, our setting may add one pricing factor (RM − E(RM ))(Rsk − E(Rsk )). It is worth
analyzing what may make it necessary to account explicitly for this additional factor within the
SDF, beyond the common cubic market return (RM − E(RM ))3 . First we note that by definition
of the tracking error
(RM − E(RM ))(Rsk − E(Rsk )) ̸= (RM − E(RM )3 ⇔ T sk ̸= 0.
35
But, even more important we have:
[
]
[
]
Cov (RM − E(RM ))(Rsk − E(Rsk )), Ri =
̸
Cov (RM − E(RM )3 , Ri
[
]
[
]
⇔ Cov (RM − E(RM ))T sk ), Ri =
̸
0 ⇔ E (RM − E(RM ))T sk Ri ̸= 0.
In other words, the tracking error, albeit uncorrelated with all asset returns, may become
correlated when multiplied by the market return. The resulting additional pricing factor has a
price proportional to the cross-sectional variance V ar(ρ) of skew tolerances across investors. In
particular, this additional factor is priced in equilibrium if and only if investors are heterogeneous
in terms of skew tolerances. We call co-cross skewness of asset i the risk quantity Cov[(RM −
E(RM ))(Rsk − E(Rsk )), Ri ] that shows up in the risk premium formula (57). The importance
of the cross-sectional variance V ar(ρ) in H(3, 2) should not come as a surprise: it matches a
similar result of Constantinides and Duffie (1996), where market incompleteness (here: tracking
error (T sk − E(T sk ))) gives rise to an additional pricing factor with a price proportional to the
variance of the distribution of heterogeneity.
As far as market incompleteness is concerned, it is worth noting that non-zero individual
kurtosis tolerances will not lead the agents to invest directly in the cubic market return but only
in its best tracking portfolio with return Rkurt corresponding to the affine regression defined in
(53):
R
kurt
= Rf +
n
∑
ξikurt (Ri − Rf ), where ξ kurt = (ξikurt )1≤i≤n = σ 2 Σ−1 d.
i=1
Here, dM = (di,M )i is the vector of market co-kurtosis of asset i introduced in definition 1,
ξ kurt defines the so-called kurtosis portfolio that is hold in equilibrium by agent s insofar as her
kurtosis tolerance differs from the average. Similarly, due to market incompleteness the agent
cannot invest into (RM − E(RM ))(Rsk − E(Rsk )) but only in the return R∗sk of the best tracking
portfolio ξ ∗sk , corresponding to its affine regression defined in (54):
R
∗sk
= Rf +
n
∑
ξi∗sk (Ri − Rf ), where ξ ∗sk = σ 2 Σ−1 c(ξ, ξ sk ).
i=1
where we used the vector of co-cross skewness c(ξ, ξ sk ) = (ci (ξ, ξ sk ))1≤i≤n with
Definition 3 The co-cross skewness of asset i is defined as
ci (ξ, ξ sk ) =
1
Cov[(RM − E(RM )(Rsk − E(Rsk )), Ri ] = Cov[(ξ ⊥ Y )(ξ sk,⊥ Y ), Yi ].
σ4
36
Similarly to (31), we show in the appendix that the equilibrium portfolio of investor s involves
a third order term σ 2 θs2 such that:
σ 2 θs2 = −
}
τs {
kurt
∗sk
(κ
−
κ̄)
ξ
−
2[ρ
(ρ
−
ρ̄)
−
V
ar(ρ)]
ξ
−
3(ρ
−
ρ̄)V
ar(R
)
ξ
.
s
s
s
s
M
τ̄ 3
(60)
Note that, by contrast with (31), this expansion term involves two departures from the mean
variance portfolio: Most important, the kurtosis portfolio leads to a portfolio adjustment when
the kurtosis tolerance of investor s is different from average. In addition to that, the co-cross
skewness portfolio ξ ∗sk is reflected in the portfolio holdings of investor s when the term ρs (ρs − ρ̄)
differs from its cross-sectional average V ar(ρ).
5.3
Co-pentosis pricing and unspanned squared skewness portfolio
return
In this subsection, we go one step further than in the former one. We want to characterize the
pricing factors which matter to describe the approximation of the risk premium (E(Ri ) − Rf )
through the first four terms σ 2 (ai2 + σai3 + σ 2 ai4 + σ 3 ai5 ) of its expansion (3). We show that
the addition of the fourth term σ 5 ai5 may lead to introduce five additional pricing factors: Most
important, we find an additional power of the market return, namely the quartic one, to go beyond
cubic pricing kernels of the previous subsection as in the representative agent model of Dittmar
(2002). However, we also find the squared skewness portfolio factor (Rsk − E(Rsk ))2 (second),
the co-cross skewness portfolio (RM − E(RM ))(R∗sk − E(R∗sk )) (third), the co-cross-kurtosis
portfolio (RM − E(RM ))(Rkurt − E(Rkurt )) (fourth) and the co-cross quadratic market-skewness
portfolio (RM − E(RM ))2 (Rsk − E(Rsk )) (fifth additional factor). These factors will be priced
in equilibrium when investors display heterogeneous skew tolerances (for a non-zero price of the
squared skewness portfolio factor), and when these skew tolerances display some cross sectional
correlation with kurtosis tolerances (for a non-zero price of the co-cross-kurtosis and the co-cross
quadratic market-skewness portfolios).
Here we extend the definitions of aggregate characteristics of preferences introduced earlier
by introducing A1 (ρ, κ), A2 (ρ, κ) as follows:
A1 (ρ, κ) =
−ρ̄ + 2ρ̄2 − 4V ar(ρ) + κ̄
−ρ̄ + ρ̄2 − 8V ar(ρ) + 3κ̄
,
A
(ρ,
κ)
=
.
2
τ̄ 4
τ̄ 4
37
(61)
The approximation E(Ri ) − Rf ≈ ai2 σ 2 + ai3 σ 3 + ai4 σ 4 + ai5 σ 5 leads to complete theorem 3
by
Theorem 6 With heterogeneous investors s = 1, . . . , S we complete the characterization of the
∑
j
equilibrium risk premium πi (σ) = E(Ri ) − Rf = ∞
j=2 aij σ in theorems 3, 5 by
τ̄ 4 σ 5 ai5 = A1 (ρ, κ)SM bi,M + A2 (ρ, κ)VM ci,M − V ar(ρ)Cov[(Rsk − E(Rsk ))2 , Ri ] − χ̄fi,M
−4 {Skew(ρ) + ρ̄V ar(ρ)} Cov[(RM − E(RM ))(R∗sk − E(R∗sk )), Ri ]
{
}
2Cov[(RM − E(RM ))(Rkurt − E(Rkurt )), Ri ]
+Cov(ρ, κ)
.
+3Cov[(RM − E(RM ))2 (Rsk − E(Rsk )), Ri ]
(62)
where the quantities A1 (ρ, κ), A2 (ρ, κ) are defined in equation (61).
Comparing this to theorem 1, i.e. with the fifth order risk premium term in the representative agent case, we see additional pricing factors due to the cross-sectional distribution of
the preference paramaters and to the cross-sectional correlation of preference parameters. The
cross-sectional distributions as well as the cross-sectional correlation of investors’ preferences are
known to be empirically relevant to understand individual investors’ portfolio choice, see, e.g.
Noussair et al. (2014).
Regarding the impact of heterogeneity on the price of different risk factors, several remarks
are in order. First, not surprisingly, the quartic market return has a non-zero price insofar as
the investors display in average a non-zero pentosis tolerance:
[u′ (qs Rf )]3 us (qs Rf )
,
χs = s
24[u′′s (qs Rf )]4
[5]
∑S
χ̄ = ∑s=1
S
qs τ s χ s
s=1 qs τs
.
Second, the cross-sectional variance V ar(ρ) of skew-tolerances scales not only the price of the
co-cross-skewness portfolio (as seen in the previous subsection) but also the price of the squared
skewness portfolio. Third, the cross-sectional covariance Cov(ρ, κ) between skew tolerances and
kurtosis tolerances scales the price of both the co-cross-kurtosis and the co-cross quadratic
market-skewness portfolios. Here, analogous to the previous definitions of cross sectional averages and variances, this covariance is defined by
∑S
qs τs (ρs − ρ̄)(κs − κ̄)
Cov(ρ, κ) = s=1 ∑S
.
s=1 qs τs
38
Fourth, the cross-sectional skewness of skew tolerances
∑S
qs τs (ρs − ρ̄)3
Skew(ρ) = s=1∑S
s=1 qs τs
scales the product of the co-cross skewness portfolio R∗sk − E[R∗sk ] with the market return.
Finally, the quantities A1 (ρ, κ) and A2 (ρ, κ) come to modify at higher orders the prices of the
market return and the skewness portfolio respectively.
Adding σ 5 ai5 of equation (62) to the approximation (57) leads us to the following SDF, up
to a constant:
Theorem 7 Up to risk premium terms of order higher than five in σ, the SDF is, up to a
constant:
H(4) = H(4, 1) + H(3, 2) + H(4, 2) + H(4, 3) + H(4, 4) + H(4, 5),
where
{
}
1 1 − 3ρ̄
A1 (ρ, κ)
(−Rf )H(4, 1) =
+
V ar(RM ) +
Skew(RM ) RM
τ̄
τ̄ 3
τ̄ 4
{
}
ρ̄
A2 (ρ, κ)
−
−
V ar(RM ) (RM − E(RM ))2
τ̄ 2
τ̄ 4
κ̄
χ̄
+ 3 (RM − E(RM ))3 − 3 (RM − E(RM ))4 .
τ̄
τ̄
The additional pricing factor H(3, 2) is defined in (59). The additional pricing factors H(4, j)
(j=2,3,4,5) are defined as
Cov(ρ, κ)
(RM − E(RM ))(Rkurt − E(Rkurt )),
τ̄ 4
Skew(ρ) + ρ̄V ar(ρ)
(−Rf )H(4, 3) = −4
(RM − E(RM ))(R∗sk − E(R∗sk )),
τ̄ 4
Cov(ρ, κ)
(−Rf )H(4, 4) = 3
(RM − E(RM ))2 (Rsk − E(Rsk )),
4
τ̄
V ar(ρ) sk
(−Rf )H(4, 5) = −
(R − E(Rsk ))2 .
τ̄ 4
(−Rf )H(4, 2) = 2
(63)
(64)
(65)
(66)
Note that H(4, 1) is a fourth order polynomial in the market return that corresponds to the
classical quartic SDF provided by representative agent models, see, e.g. Dittmar (2002). As
usual, the four pricing factors RM , (RM − E(RM ))2 , (RM − E(RM ))3 and (RM − E(RM ))4 have
39
prices proportional to risk aversion, skew tolerance, kurtosis tolerance and pentosis tolerance,
respectively.
In comparison to the cubic SDF (58), note that adding the expansion term σ 5 a3 has led us to
adding two correction terms. First, a further correction by A1 (ρ, κ)/τ̄ 4 SM to the standard weight
(1/τ̄ ) of the market return; note this correction is in addition to the earlier term (1 − 3ρ̄)VM /τ̄ 3
that showed up in (58) as a correction to the quadratic SDF of (21). Second, a correction
−A2 (ρ, κ)/τ̄ 4 VM to the standard weight (ρ̄/τ̄ 2 ) of the squared market return. Note that both
correction terms are of lower order in small noise expansions since VM is proportional to σ 2 and
SM is proportional to σ 3 .
Similar to H(3, 2) these pricing factors embody a form of market incompleteness, here the
impossibility to hedge the squared market return, the cubic market return as well as the cross
product between market return and the return on the skewness portfolio. Ultimately, the SDF
will be a polynomial of all these factors at various powers; note that only the powers that appear
through H(3, 2) contribute to fourth moments (kurtosis pricing), while only the terms in H(4, j)
(j=2,3,4,5) lead to fifth moments (pentosis pricing).
The additional factors H(3, 2) and H(4, j) (j=2,3,4,5), defined in equations (63-63) above,
will contribute to pricing whenever there is some heterogeneity in preferences for higher order
moments (V ar(ρ), Cov(ρ, κ) and Skew(ρ)). In particular, note that even in a world with heterogeneous risk averse investors that have only non-vanishing skew tolerances, i.e. investors with
κs = χs = 0 for all s = 1, . . . , S, the additional factors H(3, 2), H(4, 3) and H(4, 5) will be priced
as long as there is heterogeneity of preferences for higher order moments with V ar(ρ) ̸= 0 or
Skew(ρ) ̸= 0.
Overall, this section shed more light on how heterogeneity of investors’ preferences do have
an impact on the spanning of the pricing kernel. The main message is threefold:
(i) In a world without investor heterogeneity, the only (possibly) relevant pricing factors are
the powers of the market return (RM − E(RM ))j , j = 1, 2, 3, 4, . . . as in the representative
agent model of Dittmar (2002).
(ii) In a world with heterogeneity, the pricing factors defined by powers of the market return
still show up with a non-zero price if and only if there is a non-zero aggregated preference
40
for the corresponding moment.
(iii) However, in a world with heterogeneity of preferences, additional pricing factors must be
considered based on tracking errors as explained above. The equilibrium prices of associated
risks can be interpreted as measures of heterogeneity of preferences for associated higher
order moments.
6
Conclusion
This paper showed a three-fund separation theorem for investors with mean-variance-skewness
preferences: agents hold the risk-free asset, the market portfolio and a skewness portfolio. In
complete markets, i.e. when the skewness portfolio is the portfolio that replicates the squared
market return, the skewness risk premium is driven by co-variation with the squared market
such that pricing is characterized by a stochastic discount factor that is a quadratic polynomial
in the market return. In incomplete markets, however, the skewness portfolio is the projection
of the squared market return on the available assets leading to a tracking error. In world with
heterogeneity of preferences, cross moments of the tracking error and the market return at various
orders contribute to pricing in addition to the skewness portfolio; in stochastic discount factors
this shows up as priced factors in addition to factors that are powers of market returns. In
a two-period extension of our baseline case, we introduce non-myopic investors: we find that
such investors hold an additional fund, called leverage portfolio, to replicate (intertemporal)
changes in the market return variance, a market volatility (leverage feedback) effect; the market
volatility then shows up as another priced factor, that is in addition to powers of market returns
and in addition to the products of tracking error portfolios with market returns at various orders.
Empirical studies of skewness risk based on stochastic discount factors that are polynomials in
the market return miss all these additional factors. Even more pricing factors should have been
introduced if, besides accomodating heterogeneity of preferences, we had allowed some investors’
heterogeneity regarding their expectations of future market volatility
41
Appendix
A
Single-period Proofs
Proof of Lemma 1. The first order conditions (2) can be rewritten as
πi (σ)E [u′ (W (σ))] + σE [u′ (W (σ))Yi ] = 0.
Differentiating w.r.t. σ gives
πi′ (σ)E [u′ (W (σ))] + πi (σ)
∂E [u′ (W (σ))]
∂E [u′ (W (σ))Yi ]
+ E [u′ (W (σ))Yi ] + σ
= 0.
∂σ
∂σ
Knowing that πi (0) = 0 we find for σ = 0 that
πi′ (0)u′ (W (0)) + u′ (W (0))E(Yi ) = 0.
We conclude πi′ (0) = 0 since u′ (W (0)) ̸= 0 and E(Yi ) = 0.
For our derivations in the appendix, we introduce the concepts of co-skewness, co-kurtosis,
and co-pentosis of asset i with a portfolio θ by
( (
)2 )
ci (θ) = Cov Yi , θ⊥ Y
,
( (
)3 )
di (θ) = Cov Yi , θ⊥ Y
,
( (
)4 )
fi (θ) = Cov Yi , θ⊥ Y
,
and we denote the co-cross-skewness and the co-cross-kurtosis with portfolios θ, θ̃ by
(
(
⊥
ci (θ, θ̃) = Cov Yi , θ Y
)(
⊥
θ̃ Y
))
(
,
(
⊥
di (θ, θ̃) = Cov Yi , θ Y
)2 (
⊥
θ̃ Y
))
.
In addition we denote the third moment of a a portfolio θ by
Sk(θ) = E
((
⊥
θ Y
)3 )
.
We will also write these as vectors as follows: c(θ) = (ci (θ))1≤i≤n , d(θ) = (di (θ))1≤i≤n , f (θ) =
(fi (θ))1≤i≤n , c(θ, θ̃) = (ci (θ, θ̃))1≤i≤n , d(θ, θ̃) = (di (θ, θ̃))1≤i≤n . Finally, for θ = ξ, we consider the
following market quantities:
c = c(ξ),
ξ0sk = Σ−1 c,
d = d(ξ), f = f (ξ),
ξ0∗sk = Σ−1 c(ξ, ξ0sk ),
42
Sk = Sk(ξ),
ξ0kurt = Σ−1 d.
A.1
Characterizing the expansion for the product of marginal utility
and net returns
This subsection derives an expansion of the product of marginal utility with the net return for an
asset. We are interested in this expansion since the first order conditions of optimal demand tell
us that the expectation of this product vanishes for all assets at the optimal demand. We will
use this representation in the next subsection A.2 to derive risk premiums in the representative
agent analysis; we will also use it in subsections A.3, A.4 to characterize the equilibrium with
heterogeneous investors.
Assuming for sake of expositional simplicity that the utility function us is analytical,
his (σ) = u′s (Ws (θs (σ))) · (Ri (σ) − Rf )
is an analytical function of the scale parameter σ with a zero constant term since Ri (0) = Rf .
Thus, we write:
his (σ) =
∞
∑
hisj σ j .
j=1
We denote r(σ) = (Ri (σ) − Rf )1≤i≤n the vector of net risky returns. We have by definition:
ri (σ) = σYi +
∞
∑
aij σ j , i = 1, ...n
(67)
j=2
We compute the series coefficients hisj by multiplying the expansion (67) with the series
expansion of u′s (Ws (θ(σ))). We first write a Taylor series around qs Rf :
)2
)
( n
( n
∑
∑
1
⊥
⊥
2
θks
rk
θks
rk + u′′′
u′s (Ws (θ)) = u′s (qs Rf ) + u′′s (qs Rf )qs
s (qs Rf )qs
2
k=1
k=1
)3
)4
( n
( n
∑
∑
1 ′′′′
1 (5)
⊥
⊥
3
4
+ us (qs Rf )qs
θks rk + us (qs Rf )qs
θks rk + . . .
6
24
k=1
k=1
(68)
Now, we deduce from (67) the series expansion of portfolio returns:
n
∑
k=1
θks (σ)rk = σ
n
∑
θks0 Yk + σ
k=1
+σ 3
n
∑
2
n
∑
(θks1 Yk + θks0 ak2 )
k=1
(θks2 Yk + θks1 ak2 + θks0 ak3 )
k=1
+σ 4
n
∑
(θks3 Yk + θks2 ak2 + θks1 ak3 + θks0 ak4 ) + . . .
k=1
43
(69)
Jointly, (68) and (69) imply:
(
u′s (Ws (θ)) = u′s (qs Rf ) + σu′′s (qs Rf )qs
+ σ2


u′′s (qs Rf )qs
( n
∑
n
∑
)
θks0 Yk
k=1
)
(θks1 Yk + θks0 ak2 )
1
+ u′′′
(qs Rf )qs2
2 s
( n
∑
)2 

θks0 Yk


k=1
k=1


∑n
θks1 ak2 + θks0 ak3 ))

 +u′′s (qs Rf )qs ( ∑k=1 (θks2 Yk +∑
n
n
2
′′′
3
+u (qs Rf )qs ( k=1 θks0 Yk ) ( k=1 θks1 Yk + θks0 ak2 )
+ σ
∑

 1 s ′′′′
3
+ 6 us (qs Rf )qs3 ( nk=1 θks0 Yk )
∑n
 ′′

us (qs Rf )qs ( ∑
(θks3 Yk + θks2
a
+
θ
a
+
θ
a
))
k2
ks1
k3
ks0
k4


k=1
∑n


n
′′′
2




θ
Y
)
(
θ
Y
+
θ
a
+
θ
a
)
ks0
k
ks2
k
ks1
k2
ks0
k3
 +us (qs Rf )qs ( ∑

k=1
k=1
2
n
1
′′′
2
4
+ 2 us (qs Rf )qs ( k=1 (θks1 Yk + θks0 ak2 ))
+ σ
+ ...
∑n


2 ∑n
1 ′′′′
3


+ u (q R )q (
θks0 Yk ) ( k=1 θks1 Yk + θks0 ak2 )




 21 s(5) s f s 4 ∑k=1

4
n
+ 24 us (qs Rf )qs ( k=1 θks0 Yk )
Multiplying this with the expansion (67) of ri and collecting terms, this now implies that:
his1 = u′s (qs Rf )Yi ,
his2 =
u′s (qs Rf )ai2
(
+
u′′s (qs Rf )qs
his3 = u′s (qs Rf )ai3 + u′′s (qs Rf )qs
n
∑
( k=1
n
∑
)
θks0 Yk Yi ,
)
θks0 Yk ai2

)
( n
)2 
( n


∑
∑
1 ′′′
′′
2
(θks1 Yk + θks0 ak2 ) + us (qs Rf )qs
+ us (qs Rf )qs
θks0 Yk
Yi ,


2
k=1
k=1
k=1
that
his4 = u′s (qs Rf )ai4 + u′′s (qs Rf )qs
(
( n
∑
(θks0 Yk )ai3 +
k=1
n
∑
)2
n
∑
)
(θks1 Yk + θks0 ak2 )ai2
k=1
1
2
+ u′′′
θks0 Yk ai2
s (qs Rf )qs
2
k=1


∑n
θks1 ak2 + θks0 ak3 ))

 +u′′s (qs Rf )qs ( ∑k=1 (θks2 Yk +∑
n
n
2
′′′
Yi ,
+ +us (qs Rf )qs ( ∑
k=1 θks1 Yk + θks0 ak2 )
k=1 θks0 Yk ) (

 1 ′′′′
3
n
3
+ 6 us (qs Rf )qs ( k=1 θks0 Yk )
44
and that
(
his5 = u′s (qs Rf )ai5 + u′′s (qs Rf )qs
n
∑
)
θks0 Yk
ai4
k=1

( n
)
( n
)2 


∑
∑
1
2
+ u′′s (qs Rf )qs
(θks1 Yk + θks0 ak2 ) + u′′′
(q
R
)q
θ
Y
a
s f s
ks0 k
s

 i3
2
k=1
k=1
 ′′

∑n
 us (qs Rf )qs ( ∑

k2 + θks0 ak3 ))
k=1 (θks2 Yk + θks1
∑a
n
n
′′′
2
θ
Y
)
(
θ
Y
+
θ
a
)
+ +us (qs Rf )qs ( ∑
a
ks0
k
ks1
k
ks0
k2
k=1
k=1
 1 ′′′′
 i2
3
n
3
+ 6 us (qs Rf )qs ( k=1 θks0 Yk )
∑n
 ′′

us (qs Rf )qs ( ∑
(θks3 Yk + θks2
a
+
θ
a
+
θ
a
))
k2
ks1
k3
ks0
k4


k=1
∑n


n
′′′
2




+u
(q
R
)q
(
θ
Y
)
(
θ
Y
+
θ
a
+
θ
a
)
s
f
ks0
k
ks2
k
ks1
k2
ks0
k3


s
s
k=1
k=1
∑
2
n
1 ′′′
2
+ + 2 us (qs Rf )qs ( ∑k=1 (θks1 Yk + θ∑
Yi .
ks0 ak2 ))


2
n
n
1 ′′′′
3


+ u (q R )q (
θks0 Yk ) ( k=1 θks1 Yk + θks0 ak2 )




 21 s(5) s f s 4 ∑k=1

4
n
+ 24 us (qs Rf )qs ( k=1 θks0 Yk )
A.2
Proof of theorem 1
This subsection uses the results of the previous subsection to prove theorem 1. The representative
investor economy can be interpreted as an economy with a single investor S = 1, utility function
us = u and demand θ = ξ, i.e. θis0 = ξi , θisj = 0, j = 1, 2, . . ..
We denote gi (σ) = u′ (W (σ))(Ri (σ) − Rf ) the product of net return of asset i and marginal
utility, evaluated with demand equal supply (θ = ξ). Assuming for sake of expositional simplicity
that the utility function u is analytical, gi (σ) is an analytical function of the scale parameter σ
with a zero constant term since Ri (0) = Rf . Thus we write:
gi (σ) =
∞
∑
gij σ j
j=1
The first-order conditions then read for i = 1, . . . , n:
E(gij ) = 0, ∀j = 1, 2, 3, ...
(70)
Our goal is to characterize the coefficients aij , j = 2, 3, .. implied by equations (70). Our analysis
in A.1 described the expansion terms hisj ; here these become the gij terms and the analysis there
can be used to describe the first five terms in the expansion of gi (σ) for each i = 1, ..., n. Recall
that E(Yi ) = 0 for i = 1, ..., n. Up to order five, the informative optimality conditions are as
follows:
45
1. E(gi2 ) = 0 gives for i = 1, ..., n:
′
′′
0 = u (qRf ) ai2 + u (qRf ) q
n
∑
ξk Cov(Yk , Yi ),
k=1
i.e. ai2 = −
u′′ (qRf ) q
1
Cov(Yi , ξ ⊥ Y ) = bi,M .
′
u (qRf )
τ
2. E(gi3 ) = 0 gives for i = 1, ..., n:
[
]
1
2
⊥
2
0 = u′ (qRf ) ai3 + u[3]
s (qRf ) q Cov Yi , (ξ Y ) ,
2
1 u[3] (qRf ) q 2
ρ
i.e. ai3 = −
ci,M = − 2 ci,M .
′
2 u (qRf )
τ
3. E(gi4 ) = 0 gives for i = 1, ..., n:
0 = u′ (qRf ) ai4 + u′′ (qRf ) q
(
+u[3] (qRf ) q 2
n
∑
( n
∑
)
ξk ak2 ai2
)k=1
ξk ak2
k=1
1
+ u[3] (qRf ) q 2 V ar(ξ ⊥ Y )ai2
2
[
] 1
[
]
Cov Yi , ξ ⊥ Y + u[4] (qRf ) q 3 Cov Yi , (ξ ⊥ Y )3 ,
6
i.e.
ai4 =
1 ⊥
ρ
ρ ⊥
κ
⊥
d
+
(ξ
a
)a
−
V
ar(ξ
Y
)a
−
2
(ξ a2 )bi,M .
i,M
2
i2
i2
τ3
τ
τ2
τ2
However, we have seen that:
ξ ⊥ a2 =
Using this and the formula ai2 =
bi,M
,
τ
ai4 =
ξ ⊥ bM
VM
=
.
τ
τ
we conclude:
κ
1
d
+
(1 − 3ρ)VM bi.M .
i,M
τ3
τ3
4. E(gi5 ) = 0 gives for i = 1, ..., n:
(
)
1
0 = u′ (qRf ) ai5 + u′′ (qRf ) q ξ ⊥ a2 ai3 + u[3] (qRf ) q 2 V ar(ξ ⊥ Y )ai3
2
)
)
)(
(
(
1
+u′′ (qRf ) q ξ ⊥ a3 ai2 + u[4] (qRf ) q 3 E(ξ ⊥ Y )3 ai2 + u[3] (qRf ) q 2 Cov ξ ⊥ Y, Yi ξ ⊥ a3
6
((
((
)2 ) (
)
)4 )
1 [4]
1
+ u (qRf ) q 3 Cov ξ ⊥ Y , Yi ξ ⊥ a2 + u[5] (qRf ) q 4 Cov ξ ⊥ Y , Yi ,
2
24
that is
VM
ρ
χ
fi,M + 2 ai3 − 2 VM ai3
4
τ
τ
τ
)
(
κ
1 ( ⊥ ) bi,M
κ
bi,M
2ρ
VM
+ ξ a3
+ 3 SM
− 2 bi,M ξ ⊥ a3 + 3 3 ci,M
.
τ
τ
τ
τ
τ
τ
τ
ai5 = −
46
As above we have used here the known identities:
ξ ⊥ bM
VM
bi,M
=
, ai2 =
.
τ
τ
τ
ξ ⊥ a2 =
If we now remember that similarly:
ξ ⊥ a3 = −
ρ
ρ
SM , ai3 = − 2 ci,M ,
2
τ
τ
we end up with
ai5 = −
A.3
3κ − ρ(1 − ρ)
κ − ρ(1 − 2ρ)
χ
fi,M +
VM ci,M +
SM bi,M .
4
4
τ
τ
τ4
Characterizing individual demand for given risk premiums
The first order conditions (26) then read for i = 1, ..., n:
E(hisj ) = 0, ∀j = 1, 2, 3, . . .
(71)
Our earlier analysis in subsection A.1 characterized the first five terms in the expansion of
his for each i = 1, ..., n, we study in our final step the expansion of the first order conditions
E(hisj ) = 0 for j = 1, ..., 5 and all i = 1, ..., n. Recall that E(Yi ) = 0 for i = 1, ..., n by definition,
such that E(his1 ) = 0 for i = 1, ..., n. Up to order five, the informative first order conditions are
as follows:
1. E(his2 ) = 0 gives for i = 1, ..., n:
0=
u′s (qs Rf )ai2
+
u′′s (qs Rf )qs
n
∑
θks0 Cov(Yk , Yi ).
k=1
In vector-matrix notation this reads
0 = u′s (qs Rf )a2 + u′′s (qs Rf )qs Σθs0 i.e. θs0 = τs Σ−1 a2 ,
where
τs = −
(72)
u′s (qs Rf )
qs u′′s (qs Rf )
stands for the relative risk tolerance.
2. The condition E(his3 ) = 0 gives for i = 1, ..., n:
(
)2 
n
∑
1
2

θks1 Cov(Yk , Yi ) + u′′′
0 = u′s (qs Rf )ai3 + u′′s (qs Rf )qs
θks0 Yk , Yi  .
s (qs Rf )qs Cov
2
k=1
k=1
n
∑
47
Using the definition of co-skewness with a portfolio that we introduced at the beginning of this
appendix, this reads in vector-matrix notation:
1
2
0 = u′s (qs Rf )a3 + u′′s (qs Rf )qs Σθs1 + u′′′
s (qs Rf )qs c(θs0 ).
2
This gives
θs1
[
]
u′′′
u′s (qs Rf ) −1
ρs
s (qs Rf )qs −1
−1
= − ′′
Σ c(θs0 ) − ′′
Σ a3 = τs Σ
c(θs0 ) 2 + a3 ,
2us (qs Rf )
us (qs Rf )qs
τs
where
(73)
u′s (qs Rf )u′′′
s (qs Rf )
ρs =
′′
2[us (qs Rf )]2
is the skew tolerance.
3. The condition E(his4 ) = 0 gives for i = 1, ..., n:
( ⊥ )
1
⊥
2
0 = u′s (qs Rf )ai4 + u′′s (qs Rf )qs (θs0
a2 )ai2 + u′′′
s (qs Rf )qs V ar θs0 Y ai2
2
( ⊥
)
[( ⊥ ) ( ⊥
) ]
′′
′′′
⊥
+us (qs Rf )qs Cov θs2 Y, Yi + us (qs Rf )qs2 Cov θs0
Y θs1 Y + θs0
a2 , Y i
[(
)3 ]
1
3
⊥
+ u′′′′
(q
R
)q
Cov
θ
Y
, Yi .
s
f
s
s0
6 s
Using the definitions of co-cross-skewness and co-kurtosis with a portfolio that we introduced at
the beginning of this appendix, this reads in vector-matrix notation:
( ⊥ )
1
2
⊥
0 = u′s (qs Rf )a4 + u′′s (qs Rf )qs (θs0
a2 )a2 + u′′′
s (qs Rf )qs V ar θs0 Y a2
2
1 ′′′′
3
2
′′′
2 ⊥
+u′′s (qs Rf )qs Σθs2 + u′′′
s (qs Rf )qs c(θs0 , θs1 ) + us (qs Rf )qs (θs0 a2 )Σθs0 + us (qs Rf )qs d(θs0 ).
6
This gives
( ⊥ )
qs ρ s
V ar θs0
Y a2
τs
qs ρ s ⊥
qs κ s
qs ρs
c(θs0 , θs1 ) + 2
θs0 a2 Σθs0 − 2 d(θs0 ),
+2
τs
τs
τs
⊥
qs Σθs2 = qs τs a4 − qs (θs0
a2 )a2 +
where
κs =
[u′s (qs Rf )]2 u′′′′
s (qs Rf )
′′
6[us (qs Rf )]3
is the kurtosis tolerance.
48
(74)
4. Finally, the condition E(his5 ) = 0 gives for i = 1, ..., n:
( ⊥ )
1
⊥
2
0 = u′s (qs Rf )ai5 + u′′s (qs Rf )qs (θs0
a2 )ai3 + u′′′
s (qs Rf )qs V ar θs0 Y ai3
2
′′
⊥
⊥
+us (qs Rf )qs (θs1 a2 + θs0 a3 )ai2
[(
[ ⊥
]
)3 ]
1 ′′′′
′′′
2
⊥
3
⊥
+us (qs Rf )qs Cov θs0 Y, θs1 Y ai2 + us (qs Rf )qs E θs0 Y
ai2
6
(
)
[
(
) ]
⊥
2
⊥
⊥
⊥
⊥
+u′′s (qs Rf )qs Cov θs3
Y, Yi + u′′′
(q
R
)q
Cov
(θ
Y
)
θ
Y
+
θ
a
+
θ
a
, Yi
s
f
2
3
s
s
s0
s2
s1
s0
[
]
( ⊥
)2
1
2
⊥
θs1 Y + θs0
a2 , Yi
+ u′′′
s (qs Rf )qs Cov
2
[(
) ]
)4 ]
[ ⊥ 2( ⊥
1
1 ′′′′
⊥
⊥
4
a2 , Yi + u[5]
Y ) θs1 Y + θs0
Y
, Yi .
Cov
θ
(q
R
)q
+ us (qs Rf )qs3 Cov (θs0
s f s
s0
2
24 s
Using the definition of co-pentosis with a portfolio that we introduced at the beginning of this
appendix, this reads in vector-matrix notation:
( ⊥ )
1
⊥
2
0 = u′s (qs Rf )a5 + u′′s (qs Rf )qs (θs0
a2 )a3 + u′′′
s (qs Rf )qs V ar θs0 Y a3
2
1 ′′′′
′′
⊥
⊥
+us (qs Rf )qs (θs1 a2 + θs0 a3 )a2 + us (qs Rf )qs3 Sk (θs0 ) a2
6
′′′
2 ⊥
′′
+us (qs Rf )qs (θs0 Σθs1 )a2 + us (qs Rf )qs Σθs3
(
( ⊥
)
)
2
⊥
+u′′′
s (qs Rf )qs c(θs0 , θs2 ) + θs1 a2 + θs0 a3 (Σθs0 )
(
)
1
2
⊥
+ u′′′
s (qs Rf )qs c(θs1 ) + 2(θs0 a2 )(Σθs1 )
2
(
)
1
1 ′′′′
⊥
a2 )c(θs0 ) + u[5]
(qs Rf )qs4 f (θs0 ),
+ us (qs Rf )qs3 d(θs0 , θs1 ) + (θs0
2
24 s
which gives
( ⊥ )
qs ρs
⊥
⊥
V ar θs0
Y a3 − qs (θs1
a2 + θs0
a3 )a2
(75)
τs
qs κ s
qs ρ s ⊥
− 2 Sk (θs0 ) a2 + 2
(θ Σθs1 )a2
τs
τs s0
)
) qs ρ s (
( ⊥
)
qs ρs (
⊥
⊥
a3 (Σθs0 ) +
a2 + θs0
a2 )(Σθs1 )
+2
c(θs0 , θs2 ) + θs1
c(θs1 ) + 2(θs0
τs
τs
)
q
χ
qs κ s (
s
s
⊥
a2 )c(θs0 ) + 3 f (θs0 ),
−3 2 d(θs0 , θs1 ) + (θs0
τs
τs
⊥
qs Σθs3 = qs τs a5 − qs (θs0
a2 )a3 +
where
[u′ (qs Rf )]3 us (qs Rf )
χs = s
24[u′′s (qs Rf )]4
[5]
is the pentosis tolerance.
49
A.4
Proof of Theorem 3: Characterizing the equilibrium allocation
with heterogeneous investors
This section uses the representation of the previous subsection to determine the market clearing
equilibrium allocation. This provides the proof of Theorem 3.
Market clearing conditions are:
S
∑
qs θs0 = S q̄ξ
S
∑
and
qs θsj = 0, j = 1, 2, 3.
s=1
s=1
1. Plugging in the value of θs0 given by (72), the first market clearing condition gives
S
∑
qs τs Σ−1 a2 = S q̄ξ, i.e. a2 =
s=1
where
1
τs
b, θs0 = ξ,
τ̄
τ̄
(76)
∑S
s=1 qs τs
τ̄ =
S q̄
.
2. Plugging in the value of θs1 given by (73), the second market clearing condition (j = 1)
gives
S
∑
s=1
[
qs τ s Σ
−1
]
ρs
c(θs0 ) 2 + a3 = 0.
τs
Equation (76) implies c(θs0 ) = τs2 /τ̄ 2 c. We deduce
a3 = −
τs
τs
ρ̄
−1
c,
θ
=
(ρ
−
ρ̄)Σ
c
=
(ρs − ρ̄)ξ0sk ,
s1
s
τ̄ 2
τ̄ 2
τ̄ 2
where
∑S
ρ̄ = ∑s=1
S
qs τs ρs
s=1 qs τs
.
3. Using the representation of θs2 given in (74), the third market clearing condition (j = 2)
gives:
)
S (
∑
( ⊥ )
qs ρs
⊥
0 = a4
qs τs −
qs (θs0 a2 )a2 −
V ar θs0 Y a2
τ
s
s=1
s=1
(
)
S
∑
qs ρs
qs ρ s ⊥
qs κ s
−
−2
c(θs0 , θs1 ) − 2
(θs0 a2 )Σθs0 + 2 d(θs0 ) .
τ
τ
τs
s
s
s=1
S
∑
Noting that
θs0 =
τs
1
1
τ3
ξ, a2 = b = Σξ, d(θs0 ) = s3 d,
τ̄
τ̄
τ̄
τ̄
50
we get
S (
∑
qs ρ s τ s
qs ρ s τ s
qs κs τs )
τs
d ,
0 = a4
qs τs −
qs 2 (ξ ⊥ Σξ)a2 − 3 2 (ξ ⊥ Σξ)a2 − 2 3 (ρs − ρ̄)c(ξ, ξ0sk ) +
τ̄
τ̄
τ̄
τ̄ 3
s=1
s=1
S
∑
that is
a4 =
where
1 − 3ρ̄ ⊥
V ar(ρ)
κ̄
sk
(ξ
Σξ)b
−
2
c(ξ,
ξ
)
+
d,
0
τ̄ 3
τ̄ 3
τ̄ 3
∑S
κ̄ =
qs τs κs
,V
∑s=1
S
s=1 qs τs
∑S
ar(ρ) =
s=1 qs τs (ρs −
∑S
s=1 qs τs
ρ̄)2
.
Based on (74) we then find that
θs2 = 3
τs
τs
τs
(ρs − ρ̄) (ξ ⊥ Σξ) ξ + 2 3 (ρs (ρs − ρ̄) − V ar(ρ)) ξ0∗sk − 3 (κs − κ̄)ξ0kurt .
3
τ̄
τ̄
τ̄
4. Using the representation of θs3 given in (75), the fourth market clearing condition (j = 3)
gives:
0 = a5
S
∑
qs τs +
s=1
+
S
∑
s=1
S
∑
S
∑
s=1
−
⊥
−qs (θs0
a2 )a3 +
( ⊥ )
qs ρs
⊥
⊥
V ar θs0
Y a3 − qs (θs1
a2 + θs0
a3 )a2
τs
q s ρs ⊥
qs κ s
Sk(θs0 )a2 + 2
(θ Σθs1 )a2
2
τs
τs s0
) qs ρs (
)
qs ρ s (
⊥
⊥
⊥
c(θs0 , θs2 ) + (θs1
a2 + θs0
a3 )(Σθs0 ) +
c(θs1 ) + 2(θs0
a2 )(Σθs1 )
τs
τs
s=1
) qs χs
qs κ s (
⊥
−3 2 d(θs0 , θs1 ) + (θs0
a2 )c(θs0 ) + 3 f (θs0 ).
τs
τs
+
2
Recalling that:
θs0 =
τs
1
ρ̄
τs
τ4
ξ, a2 = Σξ, a3 = − 2 c, θs1 = 2 (ρs − ρ̄)Σ−1 c, f (θs0 ) = s4 f, ξ ⊥ c = Sk,
τ̄
τ̄
τ̄
τ̄
τ̄
51
we get
0 = a5
+
S
∑
s=1
S
∑
s=1
qs τs +
S
∑
qs τs
s=1
−
τ̄ 4
(ξ ⊥ Σξ)ρ̄c −
qs τ s ρ s ⊥
qs τs (ρs − 2ρ̄)
(ξ Σξ)ρ̄c −
(Sk · b)
4
τ̄
τ̄ 4
qs τ s κ s
qs τs ρs (ρs − ρ̄) ⊥
qs τs ρs
(Sk · b) + 2
(ξ c)b + 6 4 (ρs − ρ̄)(ξ ⊥ Σξ)c
4
4
τ̄
τ̄
τ̄
S
∑
qs τs ρs
qs τs ρs
+
4 4 (ρs (ρs − ρ̄) − V ar(ρ)) c(ξ, ξ0∗sk ) − 2 4 (κs − κ̄) c(ξ, ξ0kurt )
τ̄
τ̄
s=1
+
+
S
∑
qs τ s
qs τs ρs
qs τs ρs
2 4 (ρs − 2ρ̄) (Sk · b) + 4 (ρs − ρ̄)2 c(ξ0sk ) + 2 4 (ρs − ρ̄) (ξ ⊥ Σξ)c
τ̄
τ̄
τ̄
s=1
S
∑
s=1
−3
qs τs κs
qs τ s κ s ⊥
qs τs χs
sk
)
−
3
(ρ
−
ρ̄)
d(ξ,
ξ
(ξ
Σξ)c
+
f,
s
0
τ̄ 4
τ̄ 4
τ̄ 4
that is
a5 =
where
−ρ̄ + 2ρ̄2 − 4V ar(ρ) + κ̄
−ρ̄ + ρ̄2 − 8V ar(ρ) + 3κ̄ ⊥
(ξ
Σξ)c
+
(Sk · b)
τ̄ 4
τ̄ 4
}
Skew(ρ) + ρ̄V ar(ρ)
Cov(ρ, κ) {
∗sk
kurt
sk
−4
c(ξ,
ξ
)
+
2c(ξ,
ξ
)
+
3d(ξ,
ξ
)
0
0
0
τ̄ 4
τ̄ 4
V ar(ρ) sk
χ̄
−
c(ξ0 ) − 4 f,
4
τ̄
τ̄
∑S
χ̄ =
qs τs χs
, Cov(ρ, κ)
∑s=1
S
s=1 qs τs
∑S
=
s=1 qs τs (ρs − ρ̄)(κs
∑S
s=1 qs τs
− κ̄)
∑S
, Skew(ρ) =
s=1 qs τs (ρs −
∑S
s=1 qs τs
(77)
ρ̄)3
.
In terms of higher moments of returns, one may want to reread formula (77) for risk premium
as:
[
]
[
]
Skew(ρ) + ρ̄V ar(ρ)
σ 5 a5 = A1 (ρ, κ)E (ξ ⊥ Y )3 Cov Y, (ξ ⊥ Y ) − 4
Cov[(ξ ⊥ Y )(ξ0∗sk,⊥ Y ), Y ]
4
τ̄
[ ⊥ ]
[ ⊥ 2 ] V ar(ρ)
[ ⊥ 4 ]
χ̄
sk,⊥ 2
+A2 (ρ, κ)V ar ξ Y Cov (ξ Y ) , Y −
Cov[(ξ
)
,
Y
]
−
Cov
(ξ Y ) , Y
0
τ̄ 4
τ̄ 4
}
Cov(ρ, κ) {
kurt,⊥
⊥
⊥
2 sk,⊥
+
2Cov[(ξ
Y
)(ξ
Y
),
Y
]
+
3Cov[(ξ
Y
)
(ξ
Y
),
Y
]
.
0
0
τ̄ 4
where A1 (ρ, κ), A1 (ρ, κ) have been defined in equation (61).
B
Online Appendix: Two-period Extension
This appendix studies the extension from a single time-period to two time-periods; it contains the
proofs of the theorems 2 (representative investor case) and 4 (heterogeneous investor case). The
52
proof of these two theorems runs largely in parallel. The first subsection derives a description of
the date 0 first order conditions of demand for given risk premiums that apply to any investor,
including the representative investor. The second subsection then uses this to describe the
equilibrium risk premium for the representative investor, while the third subsection describes
the equilibrium for heterogeneous investors.
With a conditional viewpoint, the individual demand schedule of any investor may depend on
her current wealth. To simplify our later derivation, we define for each investor s three dummy
functions fs , gs , fes by setting
1
fs (ws ) ≡ ws θs0
,
(78)
1
gs (ws ) ≡ ws θs1
,
(79)
1,⊥ 1
fes (ws ) ≡ ws θs0
a2 ,
(80)
1
1
for a given functional form of the expansion terms θs0
, θs1
, and a12 (as functions of date 1) wealth
of investor s. Note that these expansion terms may depend on the wealth of investors other than
s; note also that they depend on the realization of the vector of random variables Y1 . However, to
simplify the exposition throughout this proof, we will not write out these dependences explicitly.
B.1
Deriving the expansion of the date 0 first order condition
As discussed in the main part of the paper, the date 0 first-order conditions are for i = 1, . . . , n
and agent s = 1, . . . , S:
[
0 = E0
∂Ws,2
(π0 (σ)
u′s (Ws,2 )
∂Ws,1
]
+ σY(1) ) .
(81)
As in the single-period analysis, we interpret the date 0 first-order conditions as a σ series of
first order conditions. To simplify our exposition, we assume that
h0s (σ) = u′s (Ws,2 )
∂Ws,2
(π0 (σ) + σY(1) )
∂Ws,1
is an analytical function of the scale parameter σ; it has a zero constant term since R(1) (0) = Rf .
Thus we write
h0s (σ)
=
∞
∑
j=1
53
h0sj σ j .
(82)
Note that equations (81, 82) imply that for all j = 1, 2, 3, ...:
[ ]
E0 h0sj = 0.
(83)
Our goal in this subsection is to characterize these expansion terms. Throughout our analysis
we use repeatedly
π0 (σ) + σY(1) = σY(1) +
∞
∑
a0j σ j ,
and π1 (σ) + σY(2) = σY(2) +
∞
∑
a1j σ j .
(84)
j=2
j=2
We note based on the series expansion of the individual demand schedule and (84) that
terminal wealth can be expressed as
(
)
Ws,2 = Ws,1 Rf + θs1,⊥ (π1 (σ) + σY(2) )
(
)
{
}
( )
1,⊥
1,⊥
1,⊥ 1
= Ws,1 Rf + σWs,1 θs0
Y(2) + σ 2 Ws,1 θs1
Y(2) + θs0
a2 + O σ 3 ,
(85)
(86)
and that date 1 wealth can be written
(
)
Ws,1 = qs Rf + θs0,⊥ (π0 (σ) + σY(1) )
(
)
{
}
( )
0,⊥
0,⊥
0,⊥ 0
2
= qs Rf + qs σ θs0 Y(1) + qs σ θs1 Y(1) + θs0 a2 + O σ 3 .
(87)
To determine the σ series expansion of the first order conditions, we derive next the σ series
expansion of the first two product terms on the left-hand side of equation (82). First, let us
derive the σ series expansion of
∂Ws,2
:
∂Ws,1
Using the definitions in equations (78-80) we can write
the investor’s terminal wealth Ws,2 as
Ws,2 = Rf,1 Ws,1 +
⊥
σY(2)
fs
{
}
( )
⊥
e
(Ws,1 ) + σ Y(2) gs (Ws,1 ) + fs (Ws,1 ) + O σ 3 .
2
(88)
This gives
{
}
( )
∂Ws,2
⊥ ′
⊥ ′
= Rf,1 + σY(2)
fs (Ws,1 ) + σ 2 Y(2)
gs (Ws,1 ) + fes′ (Ws,1 ) + O σ 3 .
∂Ws,1
For further calculations below we will need the σ series expansion of
∂Ws,2
.
∂Ws,1
with a series expansion of the three dummy functions around qs Rf :
fs′ (Ws,1 ) = fs′ (qs Rf ) + fs′′ (qs Rf )(Ws,1 − qs Rf ) + . . . ,
gs′ (Ws,1 ) = gs′ (qs Rf ) + gs′′ (qs Rf )(Ws,1 − qs Rf ) + . . . ,
fes′ (Ws,1 ) = fes′ (qs Rf ) + fes′′ (qs Rf )(Ws,1 − qs Rf ) + . . . .
54
(89)
For this, we start
Based on this and equation (87) we can now write out the σ series expansion of (89) as:
(
) }
{
0,⊥
⊥ ′′
( )
Y
(q
R
)q
θ
Y
f
∂Ws,2
s f s
(1)
s0
(2) s
⊥ ′
= Rf + σY(2)
fs (qs Rf ) + σ 2
+ O σ3
(90)
⊥ ′
∂Ws,1
+Y(2)
gs (qs Rf ) + fes′ (qs Rf )
Next, let us derive the σ series expansion of marginal utility derived from terminal wealth
u′s (Ws,2 ). Analogous to the derivation of (90), it can be shown that terminal wealth (88) can be
written as
{
⊥
Ws,2 = Rf Ws,1 + σY(2)
fs (qs Rf ) + σ 2
(
) }
0,⊥
⊥ ′
( )
fs (qs Rf ) qs θs0
Y(1)
Y(2)
+ O σ3 .
+Y ⊥ gs (qs Rf ) + fes (qs Rf )
(91)
(2)
Equation (91) jointly with (87) gives:
Ws,2
{
(
)}
0,⊥
⊥
+ σ Y(2) fs (qs Rf ) + qs Rf θs0 Y(1)
=
(92)


(
)
0,⊥
⊥
 Y ⊥ fs′ (qs Rf ) qs θs0
Y(1) + Y(2)
gs (qs Rf ) + fes (qs Rf ) 
( 3)
(2)
2
{
}
+σ
+
O
σ .
0,⊥
0,⊥ 0


+qs Rf θs1
Y(1) + θs0
a2
qs Rf2
Finally, we exploit (92) and perform a Taylor expansion of u′s (Ws,2 ) around qs Rf2 :
u′s (Ws,2 )
(
))
(
)
(
)( ⊥
0,⊥
′
2
′′
2
= us qs Rf + σus qs Rf Y(2) fs (qs Rf ) + qs Rf θs0 Y(1)


(
)]2
(
)[ ⊥
0,⊥
′′′
2




u
q
R
Y
f
(q
R
)
+
q
R
θ
Y
s f
s f
s f
(1)
s0
s


(2) s



(
)

1 2
0,⊥
⊥
′
⊥
.
+ σ
Y f (q R ) q θ Y
+ Y(2) gs (qs Rf )
 +2u′′ (qs R2 )  (2) s s f s s0 { (1)

2 
}  
s
f


0,⊥
0,⊥ 0


+fes (qs Rf ) + qs Rf θs1
Y(1) + θs0
a2
(93)
Now that we have a characterization of the σ series expansion of the first two product terms
on the left-hand side of equation (82), we put these together with (84). First, we note that
equation (90) jointly with (84) gives
∂Ws,2
(π0 (σ) + σY(1) )
(94)
∂Ws,1
(
)
{( ⊥ ′
)
}
= σ Rf Y(1) + σ 2 Y(2)
fs (qs Rf ) Y(1) + Rf a02
(95)

 (
)
(
)
0,⊥
⊥ ′
 Y ⊥ fs′′ (qs Rf )qs θs0
Y(1) + Y(2)
gs (qs Rf ) + fes′ (qs Rf ) Y(1) 
( )
(2)
3
)
(
+ O σ4
+σ
⊥ ′


+ Y(2)
fs (qs Rf ) a02 + Rf a03
55
Multiplying (95) and (93), we can characterize the terms in equation (82):
(
) (
)
h0s1 = Rf Y(1) u′s qs Rf2
))
(
)( ⊥
(
0,⊥
Y(1) (Rf Y(1) )
h0s2 = u′′s qs Rf2 Y(2)
fs (qs Rf ) + qs Rf θs0
(
) {( ⊥ ′
)
}
+u′s qs Rf2
Y(2) fs (qs Rf ) Y(1) + Rf a02


)]2
(
(
)[ ⊥
0,⊥
′′′
2




Y
u
q
R
Y
f
(q
R
)
+
q
R
θ
s f
s f
s f
(1)
s0
s


(2) s


(

)

(
)
1
0
0,⊥
⊥ ′
⊥
hs3 = Rf Y(1)
(
) Y(2) fs (qs Rf ) qs θs0 Y(1) + Y(2) gs (qs Rf )
2
{
}  


+2u′′s qs Rf2 


0,⊥
0,⊥ 0


e
+fs (qs Rf ) + qs Rf θs1 Y(1) + θs0 a2
(
))
{( ⊥ ′
)
} (
)( ⊥
0,⊥
+ Y(2)
fs (qs Rf ) Y(1) + Rf a02 u′′s qs Rf2 Y(2)
fs (qs Rf ) + qs Rf θs0
Y(1)
 (

)
( ⊥
)
′
⊥ ′′
⊥ ′
e

Y(2) fs (qs Rf )qs θs0,0 Y(1) + Y(2) gs (qs Rf ) + fs (qs Rf ) Y(1) 
(
)
(
)
+u′s qs Rf2
⊥ ′


+ Y(2)
fs (qs Rf ) a02 + Rf a03
For further analysis we note that fs , fs′ , fs′′ , gs , f˜s depend on Y(1) but do not depend on Y(2) .
By definition, E1 [Y(2) |Y(1) ] = 0 and E0 [Y(1) ] = 0, which implies E0 [h0s1 ] = 0 and
)]
[
)]
[
(
( ⊥
) ( 0,⊥
0,⊥ 0
⊥
= 0,
E Y(1) Y(2)
fs (qs Rf ) θs0
Y(1) = E[Y(2)
gs (qs Rf ) Y(1) ] = E Y(1) θs0
a2
[(
(
)]
[
]
)
0,⊥
0,⊥
⊥ ′
⊥
⊥
E Y(2)
fs (qs Rf ) Y(1) θs0
Y(1) = E[Y(2)
fs (qs Rf ) a02 ] = E Y(2)
(θs0
Y(1) ) = 0,
[
(
)
]
[ ⊥ ′
]
0,⊥
⊥ ′′
⊥ ′
E Y(2)
fs (qs Rf )qs θs0
Y(1) Y(1) = E Y(2)
gs (qs Rf )Y(1) = E[Y(2)
fs (qs Rf ) a02 ] = 0.
Up to order three, the informative first-order conditions (83) are then as follows:
1. E0 [h0s2 ] = 0 is equivalent to
(
)
(
)
0
0 = u′′s qs Rf2 qs Rf2 Σ0 θs0
+ u′s qs Rf2 Rf a02 .
(96)
We recall that τs,0 , defined in equation (46), stands for the (date 0) risk-tolerance of investor
(
)
s, then divide equation (96) by −u′s qs Rf2 Rf and then find that
0=
1
τs,0
0
Σ0 θs0
− a02 .
(97)
2. Next we analyze E0 [h0s3 ] = 0; it is equivalent to
((
))
( [
]
)2
( ⊥
)2
)
1 ′′′ (
0,⊥
2 2
2
θs0 Y(1) , Y(1)
u qs Rf Rf E0 Y(2) fs (qs Rf ) Y(1) + qs Rf Cov0
2 s
((
(
)
)
)
(
)
(
)
0,⊥
+u′′s qs Rf2 Rf Cov0 fes (qs Rf ) , Y(1) + u′′s qs Rf2 qs Rf2 Cov0 θs1
Y(1) , Y(1)
)( ⊥ ′
)
]
(
) [( ⊥
fs (qs Rf ) Y(2)
fs (qs Rf ) Y(1)
+u′′s qs Rf2 E0 Y(2)
)
(
)
(
)
(
(98)
+u′s qs Rf2 Cov0 fes′ (qs Rf ) , Y(1) + u′s qs Rf2 Rf a03 . = 0
56
We recall that ρs,0 , defined in equation (47), stands for the (date 0) skew-tolerance of agent
(
)
s, then divide equation (98) by −u′s qs Rf2 Rf and use the definitions of τs,0 , ρs,0 to get:
((
)
] ρ
)2
[(
)2
ρs,0 1
s,0
0,⊥
⊥ ⊥
0 = − 2 2 2 E0 Y(2) fs (qs Rf ) Y(1) − 2 Cov0
θs0 Y(1) , Y(1)
τs,0 qs Rf
τs,0
(
)
((
)
)
1
1
0,⊥
e
+
Cov0 fs (qs Rf ) , Y(1) +
Cov0 θs1 Y(1) , Y(1)
τs,0 qs Rf
τs,0
[( ⊥ ⊥
)( ⊥ ′
)
]
1
+
E
Y
f
(q
R
)
Y
f
(q
R
)
Y
0
s
f
s
f
(1)
(2)
s
(2)
s
τs,0 qs Rf2
(
)
1
′
e
− Cov0 fs (qs Rf ) , Y(1) − a03 .
(99)
Rf
B.2
Representative Agent Risk Premium
Based on our single-period analysis, but with a date 1 conditional viewpoint we know the risk
premium up to order 3:
a12 =
1 1
ρ1
bM , and a13 = − 2 c1
τ1
τ1
where we recall that
τ1 (w) = −
u′ (wRf ) u′′′ (wRf )
u′ (wRf )
,
ρ
(w)
=
.
1
wu′′ (wRf )
2 [u′′ (wRf )]2
We also recall that the representative agent’s date 1 demand is given through the aggregate
supply at date 1, i.e.
θ01 = ξ and θ11 = 0.
This implies for the functions in equation (78-80):
w
f (w) = wξ, f ′ (w) = ξ, g (w) = 0, fe(w) = wξ ⊥ a12 = ξ ⊥ b1M .
τ1
We recall the definitions of τ0 and of ρ0 ,
(
(
)
(
)
)
u′ qRf2 u′′′ qRf2
u′ qRf2
(
) and ρ0 =
τ0 = −
[ (
)]2 ,
qRf u′′ qRf2
2 u′′ qRf2
and that the representative agent’s date 1 demand is given through the aggregate supply at date
0, i.e.
θ00 = ξ and θ10 = 0.
57
(100)
Up to order three, the informative first-order conditions (83) are characterized through equations (97, 99). Here we apply them with a single representative agent us = u, τs,0 = τ0 , ρs,0 = ρ0
and the supply (100):
1. Based on (97), E0 [h0s2 ] = 0 gives
a02 =
1
1
Σ0 ξ = b0M .
τ0
τ0
2. Based on (99), E0 [h0s3 ] = 0 and noting that ξ ⊥ Σ1 ξ = ξ ⊥ b1M , we get:
((
)
[ ⊥
] ρ0
)2
ρ0
⊥
Cov
Cov
ξ
Σ
ξ,
Y
ξ
Y
−
,
Y
0
1
0
(1)
(1)
(1)
τ02
τ02
(
)
(
)
[
]
1
1
1
1
Cov0 ξ ⊥ Σ1 ξ, Y(1) −
Cov0 fe′ (qRf ) , Y(1) .
+
+
τ0 τ1 (qRf ) Rf
Rf
a03 = −
Using the product rule we calculate that
(
)
1
∂τ
1
1
′
⊥
1
fe (qRf ) = ξ bM
− qRf
(qRf ) 2
.
τ1 (qRf )
∂w
τ1 (qRf )
Finally, we then conclude from this and noting τ1 (qRf ) = τ0 , that
((
)
[ ⊥
] ρ0
)2
ρ0
⊥
Cov
ξ
Σ
ξ,
Y
−
Cov
ξ
Y
,
Y
0
1
0
(1)
(1)
(1)
τ02
τ02
[
] γ0
[
]
1
+ 2 Cov0 ξ ⊥ Σ1 ξ, Y(1) + 2 Cov0 ξ ⊥ Σ1 ξ, Y(1)
τ0
τ0
((
)
[
] ρ0
)2
1 − ρ0 + γ0
⊥
⊥
Cov0 ξ Σ1 ξ, Y(1) − 2 Cov0 ξ Y(1) , Y(1) ,
=
τ02
τ0
a03 = −
(101)
(102)
(103)
where we recall that
γ0 = q
B.3
B.3.1
∂τ1
(qRf ).
∂w
Heterogeneous Agent Equilibrium
The date 1 (conditional) demand schedule
Based on our single-period analysis, but with a date 1 conditional viewpoint we know the equilibrium up to order 3:
τs,1
1 1
1
=
bM , θs0
ξ,
τ̄1
τ̄1
τs,1
ρ̄1
1
sk
= 2 (ρs,1 − ρ̄1 ) ξ0,1
,
= − 2 c1 , θs1
τ̄1
τ̄1
a12 =
(104)
a13
(105)
58
where
u′s (ws Rf ) u′′′
u′s (ws Rf )
s (ws Rf )
, ρs,1 (ws ) =
,
τs,1 (ws ) = −
′′
′′
ws us (ws Rf )
2 [us (ws Rf )]2
∑S
∑S
ws ρs,1 (ws )τs,1 (ws )
s ws τs,1 (ws )
τ̄1 =
, ρ̄1 = s ∑S
,
∑S
s ws
s ws τs,1 (ws )
sk
ξ0,1
= Σ−1
1 c1 .
Note that
where we recall that
τs,0
τs,1 (qs Rf ) = τs,0 ,
(106)
(
)
u′s qs Rf2
)
(
=−
Rf qs u′′s qs Rf2
(107)
stands for the (date 0) relative risk-tolerance. The above representation of the date 1 conditional
equilibrium implies
fs (ws ) = ws
B.3.2
τs,1
τs,1
τs,1
sk
ξ, gs (ws ) = ws 2 (ρs,1 − ρ̄1 )ξ0,1
, and fes (ws ) = ws 2 (ξ ⊥ b1M ).
τ̄1
τ̄1
τ̄1
(108)
The date 0 demand schedule knowing the functional form of date 1 equilibrium
Up to order three, the informative first-order conditions (83) are as follows:
1. Based on equation (97), E0 [h0s2 ] = 0 is equivalent to
0
0
θs0
= τs,0 Σ−1
0 a2 ,
(109)
where we recall τs,0 as in equation (107).
2. Next we analyze E0 [h0s3 ] = 0. Based on equation (99), using the representation in (108)
and noting that ξ ⊥ Σξ = ξ ⊥ b1M we find
((
)
)2
ρs,0
ρs,0
0,⊥
⊥ 1
Cov0 (ξ Σ ξ, Y1 ) − 2 Cov0
θs0 Y(1) , Y(1)
0 = −
(τ̄1 (qs Rf ))2
τs,0
1
1
0,⊥
Cov0 (ξ ⊥ Σ1 ξ, Y1 ) +
Σ0 θs1
+
2
(τ̄1 (qs Rf ))
τs,0
 ( τ )

∂ ws τ̄s,1
1
1

+
(qs Rf ) Cov0 (ξ ⊥ Σ1 ξ, Y1 )
Rf τ̄1 (qs Rf )
∂ws
 ( τ )

s,1
∂
w
2
s τ̄
1 
1
(qs Rf ) Cov0 (ξ ⊥ Σ1 ξ, Y1 ) − a03 .
−
Rf
∂ws
59
(110)
B.3.3
Date 0 market clearing conditions knowing the functional form of date 1
equilibrium
We first determine the equilibrium at order 0. We multiply (109) by qs , take the sum for
∑
∑
0
0
s = 1, ..., S and use the market clearing condition for (θs0
)s , i.e. Ss=1 qs θs0
= ξ Ss=1 qs , to find
( S )
S
S
∑
∑
∑
0
0
Σ0 ξ
qs = Σ 0
qs θs0 − a2
qs τs,0 = 0
s=1
s=1
s=1
Now, we have
a02 =
1
1
Σ0 ξ = b0M ,
τ̄0
τ̄0
where we recall
∑S
τ̄0 =
s=1
∑
S
qs τs,0
s=1 qs
.
We replace a02 in (109) to get
0
θs0
=
τs,0
ξ.
τ̄0
Next, we determine the equilibrium at order 1. Using the product rule, we calculate
)
(
(
)
τs,1
∂ τ̄1
∂
w
∂ ws ττ̄s,1
2
s
τ̄1
1
τs,1
∂ws
1
=
− 2 ws
.
∂ws
τ̄1
∂ws
τ̄1
τ̄1
(111)
(112)
Evaluating this at qs Rf and using equation (106) gives
 ( τ )

 ( τ )

s,1
s,1
∂
w
∂
w
2
s
s
τ̄1
τ̄1
1 
1

(qs Rf ) −
(qs Rf )
Rf τ̄1 (qs Rf )
∂ws
Rf
∂ws
= qs
τs,1 (qs Rf ) ∂ τ̄1
γs,0
(qs Rf ) =
,
3
(τ̄1 (qs Rf )) ∂ws
(τ̄1 (qs Rf ))2
(113)
where we recall that
γs,0 = qs
τs,0 ∂ τ̄1
(qs Rf ).
τ̄0 ∂ws
Using equation (113) and equation (111) as well as noting τ̄1 (qRf ) = τ̄0 , we get:
((
)
( ⊥
) ρs,0
)2
1
ρs,0 − 1 − γs,0
0,⊥
⊥
ξ
Σ
ξ,
Y
+
ξ
Y
,
Y
−
Cov
Cov
Σ0 θs1
+ a03 = 0.
1
0
0
(1)
(1)
(1)
2
2
τ̄0
τ̄0
τs,0
(114)
0
based on the risk premium
This equation allows us to determine individual date 0 demand θs1
a03 ; in a second step we would then infer the risk premium that clears the market and in a final
step determine the equilibrium demand. However, the structure of the representation (114)
60
permits a shortcut to infer the equilibrium risk premium. For this, we multiply (114) by qs τs,0
and take the sum for s = 1, ..., S
∑S
∑S
∑S
( ⊥
)
s=1 qs τs,0 ρs,0 −
s=1 qs τs,0 +
s=1 qs τs,0 γs,0
Cov
ξ
Σ
ξ,
Y
0
1
(1)
τ̄02
( S
)
∑S
S
((
)
∑
∑
)
q
τ
ρ
2
s
s,0
s,0
⊥
0⊥
s=1
Cov0 ξ Y(1) , Y(1) − Σ0
+
qs θs1 +
qs τs,0 a03 = 0.
τ̄02
s=1
s=1
S
∑
0
We recall the market clearing conditions for {θs1
}, i.e.
∑S
s=1 qs τs,0 and get
a03
(115)
(116)
0
qs θs1
= 0, divide equation (116) by
s=1
((
)
( ⊥
) ρ̄0
)2
ρ̄0 − 1 − γ̄0
⊥
= −
Cov0 ξ Σ1 ξ, Y(1) − 2 Cov0 ξ Y(1) , Y(1) ,
τ̄02
τ̄0
(117)
where we recall that γ̄0 and ρ̄0 are given as
S
∑
γ̄0 =
S
∑
γs,0 qs τs,0
s=1
S
∑
, ρ̄0 =
qs τs,0
s=1
ρs,0 qs τs,0
s=1
S
∑
.
qs τs,0
s=1
Now, we multiply (114) by τs,0 , and replace (117) to find
((
)
( ⊥
) τs,0 ρs,0
)2
τs,0 ρs,0 − τs,0 − τs,0 γs,0
⊥
0
Cov
ξ
Σ
ξ,
Y
+
Cov
ξ
Y
,
Y
0
1
0
(1)
(1)
(1) + τs,0 a3
τ̄02
τ̄02
{
}
(
)
τs,0 (ρs,0 − ρ̄0 ) τs,0 (γs,0 − γ̄0 )
=
−
Cov0 ξ ⊥ Σ1 ξ, Y(1)
(118)
2
2
τ̄0
τ̄0
((
)
)2
τs,0 (ρs,0 − ρ̄0 )
⊥
+
Cov0 ξ Y(1) , Y(1)
τ̄02
(
)
[
]
0
Since Cov0 ξ1⊥ Σ1 ξ1 , Y(1) = V ar1 ξ1⊥ Y(2) , we find the stated representation of θs1
.
0
Σ0 θs1
=
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